I had this problem with the basic, free limewire and I switched to Pro thinking that maybe that would fix it, but the same thing keeps happening:
Whenever I click on a song or video to download, it goes to the download box as "queued" even if nothing else is downloading. The song stays "queued" and never downloads, no matter what I try. Everything else seems fine. The connection is good, the songs are coming up in searches and everything, I just can't get any of them to download.
Can someone help, please?

In LW tools > options > downloads. Make sure that your maximum downloads aren't set at zero.

Otherwise, delete LW's preferences folder when LW's properly closed. See point 3 here & also point 2 if you've used older versions of LW in the past (there could be a folder in either location).
Fixes for LimeWire
This will kill all of your settings within options, you'll have to go back through afterwards & reset things. If you're not sure of your settings, go through LW's tools > options & write them all down, so that you can reset things later.

If that doesn't help, make sure that you're giving LW some time to settle down after starting up (10 mins or so) & also make sure that you're not trying to download too many files at once.

Ah, thank you! It was the maximum downloads.
I kind of wish I'd known that before I spent the money on pro... Hah. But anyway, thanks very much.