How many bits of precision are often used in fixed point DSP?
Almost all such programming is done in C, which means that 16 bits (shorts) and 32 bits (ints) make sense.  I doubt it will surprise you that most chips support those bit widths, with 16 bits being a particular favorite because it is wide enough to be useful and narrow enough to be power efficient, small, and fast.

I believe that there are a few chips that operate on 12 bits, but that is much rarer than the 16 bit chips.