I need to replace Shimano Mountain LX Biopace Chainrings with 48/38/28 110 bcd and am having difficulty finding replacement chainrings, particularly the center chainring 38T. It appears Mountain LX is rare and I would like to know what other Shimano (Deore) or other chainrings or complete cranksets are compatible.
You cannot replace Biopace unless you find some antiques. Biopace ended 20 years ago.
Other chainrings are all compatible if the BCD is correct. The only issue is the ramps or pins that are meant to improve shifting performance. The manufacturer will always tell you to match everything up exactly but that is their sales pitch. Realistically having something ramped on the middle ring is a good idea and maybe on the big ring too. Ramps on a granny (smallest) ring don't make any difference. If you are really worried about compatibility get something similar (XT, STX, Deore...) with ramps that is from Shimano and you will be just fine.
Any square taper crankset which allows clearance for the frame and similar chainline will work fine. Actually buying a complete crankset with all the rings included is a lot cheaper than buying individual rings. The extreme example is a low end shimano riveted crankset for less than $40 vs the typical budget of $1/tooth for chainrings.
TA makes a 38t 110 bcd chainring.
For whatever reason, it's generally easier to find odd size parts on the European Web sites. Wiggle is generally a good source.
Vuelta chainrings are generally more easily available in the USA, but aren't as nice as the TA ones.
If you swap the 38t for a 36t, then you should have no problem finding standard round replacement rings. Those old Deore cranks are pretty nice, just upgrade it with a new set of rings. It's absolutely no problem to use 9spd chainrings on those cranks with 7 or 8 spd chain.
Unfortunately, Biopace components haven't been made for years (and other non-round chainrings aren't the same (e.g. Sugino Cycloid, Osymmetric)) and they've become somewhat rare. However, you can sometimes find them on eBay (listing claims Biopace, but theres no picture). Some people have stuck normal chainrings on a biopace crankset though (just need to match the BCD and number of bolts), and this is a significantly cheaper option.
As for replacement cranksets, you have just a plain old JIS square taper bottom bracket. So if you wanted a new crankset, match it to that type of bottom bracket. They're still made at the mid-lower levels of components (e.g. Shimano Alivio M430). Or you could switch the BB with another of appropriate size and get some other crankset.