I know there are plenty of words that use the -st ending: wouldst, whilst, unbeknownst, etc. but I'm not really sure what it means to add an -st suffix to a word. What does it mean to add the suffix? How can I tell what words can take the -st suffix? Are there any modern words that can take the suffix and not have people look at me in a strange way?
Long long ago in a galaxy far far away...uh, well, not really.
English used to have a more complex grammar than it does presently. It is a Germanic tongue and so retains a touch of German in old, not so much used, forms.
The -st you refer to are from the old second person singular. Wouldst:
While "wouldst" is uncommmon, I have heard "unbeknownst" used much more commonly: "Unbeknownst to me, she was married." This would likely not sound odd at all to me, just perhaps a touch formal.
Another one that is completely disused is "durst", which is the second person singular of "to dare".
ETA: Originally, I had included a couple of paragraphs about "whilst" as a verb, but a comment by @siride (that I didn't examine closely until a year after posting this answer) clued me into the fact that I was wrong about "whilst". I refer you to the comments below for more data on that.
For unbeknown vs. unbeknownst: http://fandom-grammar.livejournal.com/39346.html which would probably explain the others also. "Whilst" I've heard British people use working with them, but everything they say sounds funny and lovely!