When it was purchased, this is what was wrong:
The glove box would not stay shut most of the time. (This has since been fixed - replaced the glove box with one from a wrecked Elantra, cost $30, took five minutes to replace.)
The rear left power lock had to be locked manually. (Still broken)
Cruise control did not work. (Still broken)
The interior plastic piece that sits in the door with the window up/down buttons on it was not attached to the door on the front right side.. wouldn't stay in. (I purchased a replacement part from the same wrecked Elantra I got my glove box from, but after installing found it's not a problem with the part, but the actual part of the door it sits in - so it remains broken)
The window up/down buttons were originally black with white graphics on them, however both had rubbed off. (Still like this.)
The center console lid was broken. (Still not fixed, have not been able to find just the lid at a junk yard yet.)
Since purchasing, this is what has gone wrong additionally:
Clutch kit, clutch master cylinder, and clutch slave cylinder all went out at 93,000 miles. $1,200.
The rear left window regulator broke around 100,000 miles, which then jammed the window motor, and therefore rendered the entire assembly broken - the window would slide up and down within the door frame freely. (This was fixed easily by getting the parts from a junk yard for $40 off another Elantra, and taking the door apart myself to fix. Took about a half hour, only needed a screwdriver.)
The little metal strips that separate the back windows from the smaller triangular windows both started rusting around 100,000 miles. (I've not addressed this, as the strips are isolated and the rust seems unlikely to spread to anywhere else on the car.)
Both CV joints had to be replaced at 105,000 miles. $400.
Valve cover gasket oil seep at 106,000 miles. $110 (this, however, should be considered normal maintenance - the valve cover gasket is a "normal" leak of cars over 100k miles.)
Passenger side trans axle seal replaced at 107,000 miles. $160. (Technically not the car's fault - the shop who did the CV joints knocked this seal during installation, damaging the gasket and creating a gear oil leak.)
Front left wheel bearing replaced at 110,000 miles. $370.
The transmission has been "going out" for about 15,000 miles - it has occasional issues, such as when it's cold out (under 30 degrees F), second gear completely disappears and you have to skip it when shifting, but only until the car's warmed up. The car is still fully drivable and on a day to day basis, there are no problems - just those little signs letting me know I should expect to buy a new transmission in the next couple years.
There is another oil leak which just recently appeared. It's not been diagnosed, but from what I can tell it's the oil pan gasket, which should be cheap to fix (under $200 or so).
The front right door lock cannot be locked manually, happened around 100k miles. This basically means there's permanent child lock on that side, since I have to use my power lock controls if the passenger wants to get out.
The clear coat started peeling around 110k miles, in a little patch on the top of the roof - despite regular washes and waxes. The rest of the body looks excellent though.
There constantly seem to be brake problems. It's been in many different shops for this, but nothing is ever solved - there is always brake noise from all wheels, and the brakes are erratic - sometimes the pedal is hard and it's very difficult to brake, other times it feels great. Regular brake system flushes, as well as regular maintenance - pads, shoes, etc. The master cylinder has already been checked, so has the caliper, and neither seem to have problems. As of now it's a mystery.
Lastly, on hot days during the first five minutes after a cold start, the car hesitates pretty badly, and the engine chugs and feels bogged down. It's been ruled out as being an engine issue, but aside from that, no one knows. It's only an occasional issue, but has been happening since a couple months after I bought it.