My violin's A-string is fraying (unravelling a bit) near the bridge. Right now, my playing is a little squeaky there but nothing very noticeable. Will this get worse or is it fine to leave the string like this? Is there a string which I can buy which is hard to fray? And how to avoid the fraying of strings in general?
You can get strings that come with a tiny moveable hollow tube that sits on the bridge and allows the string to move as it needs through it, thus helping prevent fraying. I would recommend changing your string sooner rather than later as it is at risk of breaking.
Perhaps the string is just badly manufactured... this happens sometimes (most often in cheaper strings, but I've also had a few faulty ones where the price should have suggested better quality).
Likely though, the bridge at least contributes a bit as well. The grooves in the bridge should be filed wide and round enough to allow the string to slip a little bit without getting damaged. Might be a good idea to ask a luthier.
In the meanwhile, when you put on a new string yourself (which you certainly should do before any performance!) you can do a little bit against new damage by dry-lubricating the groove with some graphite from a pencil.
In addition to the previous answers, I would say that no matter how correctly the violin is set up, wound strings will eventually fray and break, usually at the bridge or nut. No string lives forever.