How can I find out what missions are available in Borderlands?

As I make my way through the main storyline, Claptrap will randomly appear in the top-right and say something like:

Greetings, Person X has more jobs for you at Place Y.

Or similar to that.

Unfortunately because I mostly play late at night and my room is adjacent to someone else my dad boards out to, I have to have the volume extremely low and I don't hear what person or where I can get the new mission.

Is there a log of what Claptrap says somewhere maybe?

Unfortunately, there is no way for you to view a log of what the claptrap says, but he will repeat often (quite annoying I've found).

Generally, there are only a few sources of quests in any given zone, and you can look up all the available missions in that area on the wiki. You'll see that most of the missions originate from bounty boards with only a few NPCs handing out some other ones.