I posted the following to this thread the other day，but then I found it was missing. I'm not sure if I had mistaken what I saw. I repost it here now, and the moderators may delete it as well if the former one had been deleted for some reason.
To begin with，you'd probably find it helpful if you read through a brief grammar of Chinese. Although a brief grammar solves only a limited number of specific problems you meet during your Chinese learning，it builds a framework of your overall knowledge of Chinese，and basing on the knowledge you'll have a different perspective on the problems you're faced.
PART OF SPEECH:
碰巧by chance: adverb(some may argue that 碰巧 is a verb, but it may not be true).
碰巧is almost always used before a verbal phrase and modifys it.
凑巧is an adjective. A Chinese adjective may serve as multiple syntactic constituents.
Example: --昨天我一伸手就抓住了一只苍蝇。厉害吧？ -- 那只是个巧合罢了/那只是巧合罢了。
A noun sometimes goes before another and modifies it. Uncommonly, you may say 巧合事件(special，sounding technical)，here 巧合 is used to distinguish the occurrence from others. You are more likely to say 巧合性的事件 or 那样的事情/事件是巧合性的. Here you are determining the nature of the occurrence.
巧合can also be an adjective, although not so typical one. 1) You may say "巧合的是，......" In the “ X的是” (，......) construction (as parenthesis, usually put at the beginning of a sentence, such as 奇怪的是，可怕的是，可笑的是)，X is usually an adjective.
2) And you may say 很巧合/非常巧合/十分巧合(地......)，......是很巧合/非常巧合/十分巧合的，太巧合了.
But you don't say (✘)不巧合, and 巧合 does not usually come before a noun. Sometimes you need “巧合性” to work adjectivally. Basing on these characters, we say 巧合is not typical as an adjective.