v2 at least had 4 modes: user mode, supervisor, IRQ and FIQ. They were encoded in the low 2 bits of r15 (which weren't otherwise needed, since the PC was always word-aligned).
The original Archimedes was intended to run a "preemptively multi tasking, multi-threaded, multi-user, written in Acorn-extended Modula2+" system called ARX. But when that wasn't ready in time, they ended up just porting the 8-bit BBC Micro MOS to the new 32-bit machines!
This is a great read about it all:
http://www.rougol.jellybaby.net/meetings/2012/PaulFellows/in...