it will be again soon. the diff section says this
Differential
Once you start making over about 80hp in one of these cars, you really need limited slip to get the most of out it, or you'll just be doing single wheelers everywhere. If you drive sensibly (no hard launches or burnouts) then your stock diff will last, Angela from Adelaide has a KE10 with a 225bhp turbo 5K and has never changed the stock 5.7" diff. Of course if you drive hard, which is really why you're doing the car up, right? You'll kill it. Nothing better bolts in, so it's time to get custom. For a drive-in-drive-out conversion, expect to pay about $1500.
If you get a diff the right width, you just need leaf spring mounts welded onto it. The pinion angle of these is pretty important, as is the strength of the welds, if you don't know about either of these things then pay someone to do it for you. Most stronger diffs are from larger cars, and are hence wider, so you might need to get some things shortened. It's easier to shorten a diff a lot than a little.
Some people use a Hilux diff, but this is known to be pretty heavy and perhaps a bit of an overkill unless you're making huge power. That being said, these are very popular diffs due to 4WD enthusiasts, so there is a large range of ratios (i've heard of a 5.7:1) and Limited Slip centers available.
A popular choice is the Borg Warner LSD diff from an R31 Skyline. These are an unequal-length housing, so you get one axle and one side shortened so it's all the same, and it should be about right for a Corolla. The Skylines had 3.6 or 3.9 ratio gears, auto Pintaras in that series had 4.1 gears, these are desireable. You can use the LSD center from anything with the same type of Borg Warner center, some examples are EB-onwards XR8 Falcons, and VN-onwards SS Commodores, they are the same LSD as the Skyline, just cheaper. You use Skyline axles because they are the same wheel stud pattern as KE30/55/70. Unfortunately, these LSDs aren't really known for their quality or for lasting a long time.
Probably the best choice is a Toyota T series diff. These came with drum brakes on everything but the Japanese AE86, and those disc brake diffs are very expensive. TRD LSD centers and 4.3 gears (although pretty bloody rare) are available for these diffs, and are a Torsen type so they work really well and last for yonks. If you use a TA22 diff, it's the same width drum-to-drum as a KE30/55 diff, and probably a pretty decent size for a KE20 with not-too-wide wheels or flared guards. If you have a KE70, a T-18 diff should bolt straight in, now just get the LSD center. There are plenty more cars that these came out of, check Matti's page.