By default, an inline definition is only valid in the current translation unit.
If the storage class is extern, the identifier has external linkage and the inline definition also provides the external definition.
If the storage class is static, the identifier has internal linkage and the inline definition is invisible in other translation units.
If the storage class is unspecified, the inline definition is only visible in the current translation unit, but the identifier still has external linkage and an external definition must be provided in a different translation unit. The compiler is free to use either the inline or the external definition if the function is called within the current translation unit.
As the compiler is free to inline (and to not inline) any function whose definition is visible in the current translation unit (and, thanks to link-time optimizations, even in different translation units, though the C standard doesn’t really account for that), for most practical purposes, there’s no difference between static and static inline function definitions.
The inline specifier (like the register storage class) is only a compiler hint, and the compiler is free to completely ignore it. Standards-compliant non-optimizing compilers only have to honor their side-effects, and optimizing compilers will do these optimizations with or without explicit hints.
inline and register are not useless, though, as they instruct the compiler to throw errors when the programmer writes code that would make the optimizations impossible: An external inline definition can’t reference identifiers with internal linkage (as these would be unavailable in a different translation unit) or define modifiable local variables with static storage duration (as these wouldn’t share state accross translation units), and you can’t take addresses of register-qualified variables.
Personally, I use the convention to mark static function definitions within headers also inline, as the main reason for putting function definitions in header files is to make them inlinable.
In general, I only use static inline function and static const object definitions in addition to extern declarations within headers.
I’ve never written an inline function with a storage class different from static.