Here's one I haven't seen mentioned: the Double Davy. I'm such a big fan, I made an animation of it.
I tested it against the Palomar, which is what I always had as my strongest knot. It was almost dead even: the Palomar broke 6 times and the Double Davy broke 4 times out of 10 tests on 6 lb fluoro. It sure is easier to tie than any other I've tried, especially with cold fingers! It's also a really compact knot, something that can be important.
For main-line to leader I use a J-knot, which is almost the same knot as the Double Davy. It's also much like a surgeon's knot with an extra across-the-knot turn, but won't slip, seems to be stronger, and doesn't put a kink in your line when fishing really light lures (or fly fishing). I learned this one from tuna charter guys in Ocean City, MD when I used to vacation there every summer. I made an animation of it as well...blue is the leader while green is the main line.
