Shortest code to throw SIGILL
PDP-11 Assembler (UNIX Sixth Edition), 1 byte
9

Instruction 9 is not a valid instruction on the PDP-11 (in octal, it would be 000011, which does not appear on the list of instructions (PDF)). The PDP-11 assembler that ships with UNIX Sixth Edition apparently echoes everything it doesn't understand into the file directly; in this case, 9 is a number, so it generates a literal instruction 9. It also has the odd property (unusual in assembly languages nowadays) that files start running from the start, so we don't need any declarations to make the program work.
You can test out the program using this emulator, although you'll have to fight with it somewhat to input the program.
Here's how things end up once you've figured out how to use the filesystem, the editor, the terminal, and similar things that you thought you already knew how to use:
% a.out
Illegal instruction -- Core dumped

I've confirmed with the documentation that this is a genuine SIGILL signal (and it even had the same signal number, 4, all the way back then!)