Unlike the situation with most of the other calculators, there is not one single shell which should always be used for the TI-82. The two shells that usually should be considered are CrASH and ACE. Both of these are available on ticalc.org, in the TI-82 assembly shells section.
CrASH was the "standard" TI-82 shell for quite a while, so most assembly programs work on it. However, it is not compatible with ROM version 19.006, so if you have this version, you will need to use ACE instead (see below). CrASH is also capable of running programs made for the older Ash shell.
This shell takes approximately 1K of memory on your calculator. To install it to your calculator, you must send a full memory backup, since system data must be modified for CrASH to run. Once you have installed CrASH on your calculator, run "prgmCRASH" and CrASH will start. Then you can select any assembly program on the calculator and run it.
After installing CrASH, two matrices called 'a' and 'b' will be on the calculator. You must not delete them. Also, don't worry about the large size one of them appears to have; it doesn't really use that much memory.
There is also a special version of CrASH, CrASH_19.006 which, as its name would suggest, will run on ROM 19.006. This version works only on ROM 19.006, and it will not run regular CrASH programs. Instead, it only runs programs that have been recompiled specifically for CrASH!19.006 (such programs don't work on regular CrASH).
ACE is a new shell, which is compatible with ROM 19.006 but still works on older ROM versions. Unfortunately, it does not run programs made for older shells. Since most programs were written long before ACE, this means that far fewer programs are compatible with ACE. Some programs for CrASH have been modified to run under ACE, but most have not. If you have a ROM version which can run CrASH, then it is probably best to use CrASH instead since there are more programs for it (but this depends on which specific programs you want to run). ACE can be installed and used in essentially the same way as CrASH. However, it does take about 3K of memory.
SNG is another new shell, which like ACE also works under ROM 19.006 as well as older ROM versions.
Ash is a somewhat older shell for the TI-82. Ash is very similar to CrASH, but CrASH has a few additional features. Many currently available assembly programs for the TI-82 are written for Ash. However, Ash programs also run under CrASH, so you don't need to install Ash for this reason. Ash will not, however, run CrASH programs, so you should use CrASH to be able to run all programs.
OS-82 is another older shell. This is one of the "original" shells along with Ash. Since it does not provide program relocation, programs for it will be somewhat less efficient. It can't run Ash or CrASH programs so recent programs can't be used with it. Furthermore, almost all programs for OS-82 have now been converted to Ash or CrASH, so there is little reason to use OS-82 now.
To find out the ROM version of your calculator, press MODE, then ALPHA, then S. The number displayed is your ROM version. To get out of this screen, press CLEAR (don't press ENTER, as this will run the self test).
You can run any of the main shells if you have ROM version 16.0, 17.0, 18.0, or 19.0. If you have version 19.006, then you can only use ACE. If you have any other version (probably very rare, as I have never heard from anybody with one) then you probably can't run any shell.
If you had ROM version 19.006, you would once have been unable to run assembly programs. However, there is now a new shell, ACE, which runs on ROM 19.006. This is described in more detail earlier in this document. Unfortunately, there are not as many programs for ACE, so having this ROM version greatly limits what you can run.
If you have a ROM version older than 16.0, then you probably have an even worse problem. I have never heard of anyone who actually has a version older than 16.0, although ticalc.org's page about the TI-82 lists some older versions. If you actually do have a calculator with such a ROM version, you should probably try to contact the shell authors about it.
If you find a ROM version newer than 19.006, then you have a similar problem. However, if a new version is released many other people will probably get it as well, so the chances of shells being updated for it is somewhat better.