The PLA (Programmable Logic Array) is one of the most common parts to go bad in a C64, and can cause a multitude of different failure modes. So if you have no idea where to start, the PLA is always a good first suspect. A typical scenario with a faulty PLA: a C64 has been sitting in the attic for years, and is returned back to use. Either it goes wonky right away, or it seems to work at first, but then malfunctions after some hours or days of use. Especially older PLAs suffer from inadequate passivation, making them vulnerable to moisture and oxidation. These chips also have a fairly high running temperature, and the heat is concentrated on a particularly small area of the chip, which may contribute to the high failure rate. The C64C models feature a new type of PLA chip that is known to be more reliable, so in those boards the PLA should not be your first suspect.
Some common symptoms include a simple black screen and flashing/rainbow colored startup screen characters. A failed PLA may get very hot. If the chip becomes hot quickly after switching on, it's likely dead. Generally if you get an incorrect startup with flashy/colorful/animating graphics, the PLA and VIC-II are prime suspects - especially if the borders are also affected. If you get a solid single-color screen such as violet/pink/orange on startup, the PLA should be your primary suspect as well. It can also cause more subtle symptoms, such as some specific games not working properly.
PLAs tend to fail spontaneously so it's hard to prove any of the following tips have any real effect on preventing a failure, but they may at least prolong the chip's lifespan:
CAUTION: When swapping PLA chips, make sure you don't mistake a SID for a PLA. They are the same size, and the chip locations vary depending on the board revision. You will likely kill either chip if you put it in the wrong socket and switch on the C64.