Any modern OS with ACPI/APM support will suppor t"soft off" power downs. Linux, Win2K and WinXP will all shutdown nicely if you tap the power button.
As mentioned, a CD-based OS is the best bet if you want "instant off". Any OS with a writable file system that utilises some sort of cache-before-write file system (ie: everything after DOS/FAT) won't like losing power.
A CD-based setup will mean your file system is read-only, and therefor safe from harm. Of course, with Linux you can also set file-systems to be read only and not require file-system checks after a hard power-off, and everything should be OK. If you ever need to update your software or ROMs, simply set your filesystems back to read/write in your /etc/fstab, modify the files, and set back to read-only.
AdvanceMAME/CD is tiny and fast to boot. Plus it's only 20-odd MegaBytes, which leaves a decent 600MB on a CD for ROMs (or 4GB+ if you go with a DVD). It means limiting yourself to using AdvanceMenu as the front-end, which seems to upset some people greatly. Personally speaking, if it shows me a screenshot and lets me run the game, functionality-wise I'm cool with it. Especially if it's light on bloat.

Building cabs for non-tech-savvy folk, a CD-booting OS means three things:
1) they can't break it when they yank the power cable out of the wall
2) there's one less volitile part in the system for them to break
3) the system is guaranteed to be the same on each reboot thanks to a read-only file-system.