How are asynchronous signal handlers executed on Linux?

These two explanations really aren't contradictory if you take into account the fact that Linux hackers tend to be confused about the difference between a thread and a process, mainly due to the historical mistake of trying to pretend threads could be implemented as processes that share memory. :-)

With that said, explanation #2 is much more detailed, complete, and correct.

As for the stack and register contents, each thread can register its own alternate signal-handling stack, and the process can choose on a per-signal basis which signals will be delivered on alternate signal-handling stacks. The interrupted context (registers, signal mask, etc.) will be saved in a ucontext_t structure on the (possibly alternate) stack for the thread, along with the trampoline return address. Signal handlers installed with the SA_SIGINFO flag are able to examine this ucontext_t structure if they like, but the only portable thing they can do with it is examine (and possibly modify) the saved signal mask. (I'm not sure if modifying it is sanctioned by the standard, but it's very useful because it allows the signal handler to atomically replace the interrupted code's signal mask upon return, for instance to leave the signal blocked so it can't happen again.)

Source #1 (Andries Brouwer) is correct for a single-threaded process. Source #2 (SCO Unix) is wrong for Linux, because Linux does not prefer threads in sigwait(2). Moshe Bar is correct about the first available thread.

Which thread gets the signal? Linux's manual pages are a good reference. A process uses clone(2) with CLONE_THREAD to create multiple threads. These threads belong to a "thread group" and share a single process ID. The manual for clone(2) says,

Signals may be sent to a thread group as a whole (i.e., aTGID) using kill(2), or to a specific thread (i.e., TID) usingtgkill(2).

Signal dispositions and actions are process-wide: if anunhandled signal is delivered to a thread, then it will affect(terminate, stop, continue, be ignored in) all members of thethread group.

Each thread has its own signal mask, as set by sigprocmask(2),but signals can be pending either: for the whole process(i.e., deliverable to any member of the thread group), whensent with kill(2); or for an individual thread, when sent withtgkill(2). A call to sigpending(2) returns a signal set thatis the union of the signals pending for the whole process andthe signals that are pending for the calling thread.

If kill(2) is used to send a signal to a thread group, and thethread group has installed a handler for the signal, then thehandler will be invoked in exactly one, arbitrarily selectedmember of the thread group that has not blocked the signal.If multiple threads in a group are waiting to accept the samesignal using sigwaitinfo(2), the kernel will arbitrarilyselect one of these threads to receive a signal sent usingkill(2).

Linux is not SCO Unix, because Linux might give the signal to any thread, even if some threads are waiting for a signal (with sigwaitinfo, sigtimedwait, or sigwait) and some threads are not. The manual for sigwaitinfo(2) warns,

In normal usage, the calling program blocks the signals in set via aprior call to sigprocmask(2) (so that the default disposition forthese signals does not occur if they become pending betweensuccessive calls to sigwaitinfo() or sigtimedwait()) and does notestablish handlers for these signals. In a multithreaded program,the signal should be blocked in all threads, in order to prevent thesignal being treated according to its default disposition in a threadother than the one calling sigwaitinfo() or sigtimedwait()).

The code to pick a thread for the signal lives in linux/kernel/signal.c (the link points to GitHub's mirror). See the functions wants_signal() and completes_signal(). The code picks the first available thread for the signal. An available thread is one that doesn't block the signal and has no other signals in its queue. The code happens to check the main thread first, then it checks the other threads in some order unknown to me. If no thread is available, then the signal is stuck until some thread unblocks the signal or empties its queue.

What happens when a thread gets the signal? If there is a signal handler, then the kernel causes the thread to call the handler. Most handlers run on the thread's stack. A handler can run on an alternate stack if the process uses sigaltstack(2) to provide the stack, and sigaction(2) with SA_ONSTACK to set the handler. The kernel pushes some things onto the chosen stack, and sets some of the thread's registers.

To run the handler, the thread must be running in userspace. If the thread is running in the kernel (perhaps for a system call or a page fault), then it does not run the handler until it goes to userspace. The kernel can interrupt some system calls, so the thread runs the handler now, without waiting for the system call to finish.

The signal handler is a C function, so the kernel obeys the architecture's convention for calling C functions. Each architecture, like arm, i386, powerpc, or sparc, has its own convention. For powerpc, to call handler(signum), the kernel sets the register r3 to signum. The kernel also sets the handler's return address to the signal trampoline. The return address goes on the stack or in a register by convention.

The kernel puts one signal trampoline in each process. This trampoline calls sigreturn(2) to restore the thread. In the kernel, sigreturn(2) reads some information (like saved registers) from the stack. The kernel had pushed this information on the stack before calling the handler. If there was an interrupted system call, the kernel might restart the call (only if the handler used SA_RESTART), or fail the call with EINTR, or return a short read or write.