I use the TSR-8 myself. It has its strengths and weaknesses.
On the plus side, it's fully microprocessor-controlled, which gives it nice, smooth tape handling and various other goodies like automatic punch-in. It also has built-in DBX noise reduction, with the additional ability to turn DBX off only on track 8 (e.g. for timecode) and it can be synchronised to another machine as a slave device.
On the minus side, it's only got two heads instead of three which makes alignment a problem. Also, heaven help you if anything does go wrong with the control system.
The '38 has the full three heads, but uses a simple CMOS logic system. It cannot do anything clever like slowing down as it approaches zero, or automatic punch-in. It can't be synchronized to any other machines (though other machines can be synced to it). I've heard that the sound quality isn't quite on par with the TSR-8, but I can't verify that. As with the TSR-8, some parts are still available (last I checked).
Both the '38 and the TSR-8 use 1/2" tape at 15 ips, which gives double the speed and double the track width over the 388. It does make them more expensive to run, though - the 388 uses 1/4" tape which can be had more cheaply.
Affordable 16-track machines give you the following options: Fostex or Tascam. Fostex made the B16, E16 and G16 - the E16 was something of a hit at the time and appeared in many project studios. Parts are pretty much nonexistent save for cannibalising other machines. If you're looking to get one, check the state of the heads very carefully as you won't be able to replace them at a sensible price. If you offered JRF or Phi Magnetronics enough money they'd probably make some, but I don't want to think about the price.
Tascam's offering was the MSR-16 (not to be confused with the MS16 which was a one-inch machine and more expensive). They turn up occasionally, but not much so parts are likely to be a problem, except where they are common to the TSR-8 or 38. Compatible guides etc might still be available from TEAC.
Is that helpful?