What Is/Are the Purpose(S) of Inline

what is/are the purpose(s) of inline?

Is inline just a recommendation for the compiler?

Yes.

7.1.2 Function specifiers

2 A function declaration (8.3.5, 9.3, 11.4) with an inline specifier declares an inline function. The inline
specifier indicates to the implementation that inline substitution of the function body at the point of call
is to be preferred to the usual function call mechanism. An implementation is not required to perform this
inline substitution at the point of call; however, even if this inline substitution is omitted, the other rules
for inline functions defined by 7.1.2 shall still be respected.

For example from MSDN:

The compiler treats the inline expansion options and keywords as suggestions. There is no guarantee that functions will be inlined. You cannot force the compiler to inline a particular function, even with the __forceinline keyword. When compiling with /clr, the compiler will not inline a function if there are security attributes applied to the function.

Note though:

3.2 One definition rule

3 [...]An inline function shall be defined in every translation unit in which it is used.

4 An inline function shall be defined in every translation unit in which it is used and shall have exactly
the same definition in every case (3.2). [ Note: a call to the inline function may be encountered before its
definition appears in the translation unit. —end note ] If the definition of a function appears in a translation
unit before its first declaration as inline, the program is ill-formed. If a function with external linkage is
declared inline in one translation unit, it shall be declared inline in all translation units in which it appears;
no diagnostic is required. An inline function with external linkage shall have the same address in all
translation units. A static local variable in an extern inline function always refers to the same object.
A string literal in the body of an extern inline function is the same object in different translation units.
[ Note: A string literal appearing in a default argument expression is not in the body of an inline function
merely because the expression is used in a function call from that inline function. —end note ] A type
defined within the body of an extern inline function is the same type in every translation unit.

[Note: Emphasis mine]

A TU is basically a set of headers plus an implementation file (.cpp) which leads to an object file.

Should it be explicitly stated when you have a small function (I
guess 1-4 instructions?)

Absolutely. Why not help the compiler help you generate less code? Usually, if the prolog/epilog part incurs more cost than having it inline force the compiler to generate them? But you must, absolutely must go through this GOTW article before getting started with inlining: GotW #33: Inline

What other benefits are there with writing inline?

namespaces can be inline too. Note that member functions defined in the class body itself are inline by default. So are implicitly generated special member functions.
Function templates cannot be defined in an implementation file (see FAQ 35.12) unless of course you provide a explicit instantiations (for all types for which the template is used -- generally a PITA IMO). See the DDJ article on Moving Templates Out of Header Files (If you are feeling weird read on this other article on the export keyword which was dropped from the standard.)

Is it needed to state inline in order to reduce the executable file
size, even though the compiler
(according to wikipedia [I know, bad
reference]) should find such functions
itself?

Again, as I said, as a good programmer, you should, when you can, help the compiler. But here's what the C++ FAQ has to offer about inline. So be wary. Not all compilers do this sort of analysis so you should read the documentation on their optimization switches. E.g: GCC does something similar:

You can also direct GCC to try to integrate all “simple enough” functions into their callers with the option -finline-functions.

Most compilers allow you to override the compiler's cost/benefit ratio analysis to some extent. The MSDN and GCC documentation is worth reading.

What is the use of the `inline` keyword in C?

Note: when I talk about .c files and .h files in this answer, I assume you have laid out your code correctly, i.e. .c files only include .h files. The distinction is that a .h file may be included in multiple translation units.

static inline void f(void) {} has no practical difference with static void f(void) {}.

In ISO C, this is correct. They are identical in behaviour (assuming you don't re-declare them differently in the same TU of course!) the only practical effect may be to cause the compiler to optimize differently.

inline void f(void) {} in C doesn't work as the C++ way. How does it work in C? What actually does extern inline void f(void); do?

This is explained by this answer and also this thread.

In ISO C and C++, you can freely use inline void f(void) {} in header files -- although for different reasons!

In ISO C, it does not provide an external definition at all. In ISO C++ it does provide an external definition; however C++ has an additional rule (which C doesn't), that if there are multiple external definitions of an inline function, then the compiler sorts it out and picks one of them.

extern inline void f(void); in a .c file in ISO C is meant to be paired with the use of inline void f(void) {} in header files. It causes the external definition of the function to be emitted in that translation unit. If you don't do this then there is no external definition, and so you may get a link error (it is unspecified whether any particular call of f links to the external definition or not).

In other words, in ISO C you can manually select where the external definition goes; or suppress external definition entirely by using static inline everywhere; but in ISO C++ the compiler chooses if and where an external definition would go.

In GNU C, things are different (more on this below).

To complicate things further, GNU C++ allows you to write static inline an extern inline in C++ code... I wouldn't like to guess on what that does exactly

I never really found a use of the inline keyword in my C programs, and when I see this keyword in other people's code, it's almost always static inline

Many coders don't know what they're doing and just put together something that appears to work. Another factor here is that the code you're looking at might have been written for GNU C, not ISO C.

In GNU C, plain inline behaves differently to ISO C. It actually emits an externally visible definition, so having a .h file with a plain inline function included from two translation units causes undefined behaviour.

So if the coder wants to supply the inline optimization hint in GNU C, then static inline is required. Since static inline works in both ISO C and GNU C, it's natural that people ended up settling for that and seeing that it appeared to work without giving errors.

, in which I see no difference with just static.

The difference is just in the intent to provide a speed-over-size optimization hint to the compiler. With modern compilers this is superfluous.

When to use inline function and when not to use it?

Avoiding the cost of a function call is only half the story.

do:

use inline instead of #define
very small functions are good candidates for inline: faster code and smaller executables (more chances to stay in the code cache)
the function is small and called very often

don't:

large functions: leads to larger executables, which significantly impairs performance regardless of the faster execution that results from the calling overhead
inline functions that are I/O bound
the function is seldom used
constructors and destructors: even when empty, the compiler generates code for them
breaking binary compatibility when developing libraries:
- inline an existing function
- change an inline function or make an inline function non-inline: prior version of the library call the old implementation

when developing a library, in order to make a class extensible in the future you should:

add non-inline virtual destructor even if the body is empty
make all constructors non-inline
write non-inline implementations of the copy constructor and assignment operator unless the class cannot be copied by value

Remember that the inline keyword is a hint to the compiler: the compiler may decide not to inline a function and it can decide to inline functions that were not marked inline in the first place. I generally avoid marking function inline (apart maybe when writing very very small functions).

About performance, the wise approach is (as always) to profile the application, then eventually inline a set of functions representing a bottleneck.

References:

To Inline or Not To Inline
[9] Inline functions
Policies/Binary Compatibility Issues With C++
GotW #33: Inline
Inline Redux
Effective C++ - Item 33: Use inlining judiciously

EDIT: Bjarne Stroustrup, The C++ Programming Language:

A function can be defined to be inline. For example:

inline int fac(int n)
{
  return (n < 2) ? 1 : n * fac(n-1);
}

The inline specifier is a hint to the compiler that it should attempt to generate code for a call of fac() inline rather than laying down the code for the function once and then calling through the usual function call mechanism. A clever compiler can generate the constant 720 for a call fac(6). The possibility of mutually recursive inline functions, inline functions that recurse or not depending on input, etc., makes it impossible to guarantee that every call of an inline function is actually inlined. The degree of cleverness of a compiler cannot be legislated, so one compiler might generate 720, another 6 * fac(5), and yet another an un-inlined call fac(6).

To make inlining possible in the absence of unusually clever compilation and linking facilities, the definition–and not just the declaration–of an inline function must be in scope (§9.2). An inline especifier does not affect the semantics of a function. In particular, an inline function still has a unique address and so has static variables (§7.1.2) of an inline function.

EDIT2: ISO-IEC 14882-1998, 7.1.2 Function specifiers

A function declaration (8.3.5, 9.3, 11.4) with an inline specifier declares an inline function. The inline specifier indicates to the implementation that inline substitution of the function body at the point of call is to be preferred to the usual function call mechanism. An implementation is not required to perform this inline substitution at the point of call; however, even if this inline substitution is omitted, the other rules for inline functions defined by 7.1.2 shall still be respected.

When should inline be used in Rust?

One limitation of the current Rust compiler is that it if you're not using LTO (Link-Time Optimization), it will never inline a function not marked #[inline] across crates. Rust uses a separate compilation model similar to C++ because LLVM's LTO implementation doesn't scale well to large projects. Therefore, small functions exposed to other crates need to be marked by hand. This isn't a great situation, and it's likely to be fixed in the future by some combination of improvements to LTO and MIR inlining.

#[inline(never)] is sometimes useful for debugging (separating a piece of code which isn't working as expected). In theory, it can be used for benchmarking, but that's usually a bad idea: turning off inlining doesn't prevent other inter-procedural optimizations like constant propagation. In terms of normal code, it can reduce codesize if you have a frequently used helper function which is only used for error handling.

#[inline(always)] is generally bad idea; if a function is big enough that the compiler won't inline it by default, it's big enough that the overhead of the call doesn't matter (and excessive inlining increases instruction cache pressure). There are exceptions, but you need performance measurements to justify it. This example is the sort of situation where it's worth considering. #[inline(always)] can also be used to improve -O0 code quality, but that's not usually worth worrying about.

When should I write the keyword 'inline' for a function/method?

Oh man, one of my pet peeves.

inline is more like static or extern than a directive telling the compiler to inline your functions. extern, static, inline are linkage directives, used almost exclusively by the linker, not the compiler.

It is said that inline hints to the compiler that you think the function should be inlined. That may have been true in 1998, but a decade later the compiler needs no such hints. Not to mention humans are usually wrong when it comes to optimizing code, so most compilers flat out ignore the 'hint'.

static - the variable/function name cannot be used in other translation units. Linker needs to make sure it doesn't accidentally use a statically defined variable/function from another translation unit.
extern - use this variable/function name in this translation unit but don't complain if it isn't defined. The linker will sort it out and make sure all the code that tried to use some extern symbol has its address.
inline - this function will be defined in multiple translation units, don't worry about it. The linker needs to make sure all translation units use a single instance of the variable/function.

Note: Generally, declaring templates inline is pointless, as they have the linkage semantics of inline already. However, explicit specialization and instantiation of templates require inline to be used.

Specific answers to your questions:

When should I write the keyword 'inline' for a function/method in C++?
Only when you want the function to be defined in a header. More exactly only when the function's definition can show up in multiple translation units. It's a good idea to define small (as in one liner) functions in the header file as it gives the compiler more information to work with while optimizing your code. It also increases compilation time.
When should I not write the keyword 'inline' for a function/method in C++?
Don't add inline just because you think your code will run faster if the compiler inlines it.
When will the compiler not know when to make a function/method 'inline'?
Generally, the compiler will be able to do this better than you. However, the compiler doesn't have the option to inline code if it doesn't have the function definition. In maximally optimized code usually all private methods are inlined whether you ask for it or not.
As an aside to prevent inlining in GCC, use __attribute__(( noinline )), and in Visual Studio, use __declspec(noinline).
Does it matter if an application is multithreaded when one writes 'inline' for a function/method?
Multithreading doesn't affect inlining in any way.

C++'s inline - how strong a hint is it for GCC and Clang/LLVM?

[Caveat: not a C++/GCC guru] You'll want to read up on inline here.

Also, this, for GCC/C99.

The extent to which
suggestions made by using the inline
function specifier are effective (C99
6.7.4).
GCC will not inline any functions if the -fno-inline option is
used or if -O0 is used. Otherwise, GCC
may still be unable to inline a
function for many reasons; the
-Winline option may be used to determine if a function has not been
inlined and why not.

So it appears that unless your compiler settings (like -fno-inline or -O0) are used, the compiler takes the hint. I can't comment on Clang/LLVM (or GCC really).'

I recommend using -Winline if this isn't a code-golf question and you need to know what's going on.

What Is/Are the Purpose(S) of Inline