I had occasion to need to tie a knot recently, and wanted to find the "right" one. I found igkt, and a few days later I was sitting around tying and retying as many different knots as I could. I'd like to offer some thoughts from the viewpoint of someone new to knots. Apologies if this all seems old hat to the more experienced members. This is what I've determined after a short period of trying to sort out the menagery. I have not learned all the knots out there yet, just the most common ones that get a lot of discussion.
My first impression is that the advantage to many of the "classic" knots is that they are easy to remember and quick to tie, but that many of them are subsumed in their primary role by other, better knots. They are important to learn because they are often reused in other knots.
Bends:
Reef "bend" is universally maligned.
Sheet bend is said to be insecure, unless tied as the double sheet bend.
The interlocked overhand knots class of bends (Zeppelin, Hunter's, Ashley's, Alpine Butterfly) are well thought of, with most people preferring Zeppelin or Alpine Butterfly, the other two being more prone to jams. The Alpine Butterfly Bend has the advantage of having an easy "round the hand" method of forming it, while Zeppelin may be marginally more secure or harder to jam.
Carrick's bend is well thought of and compact, but may be easier to tie incorrectly and end up with something that is not a Carrick's bend, and cannot be easily determined to be incorrect by inspection. The Alpine Butterfly and Zeppelin bends both have distinct "looks" so you can tell if you did something wrong.
Hitches:
All the "classic" hitches (two half hitches, clove hitch, etc.) seem problematic as standalone hitches, with the possible exception of the slipped buntline when a compact hitch is desired.
The notableknotindex website advocates the Sailor's Hitch, which seems ideal for hitching to a large diameter round object, like a pole or a rope.
The Rolling, or Magnus, Hitch is quick and easy for lateral pulling, but comes in three variations. The Gripping Sailor's Hitch seems better in every other way.
The Taut-Line Hitch (also in three variations, as it uses the Rolling Hitch) is an easy adjustable loop hitch, but subsumed by the Adjustable Grip Hitch.
The Trucker's "Hitch" seems useful, but seems more of a technique than an actual knot; the loop(s) used can be formed in any number of ways.
Loops in the Bight:
The Alpine Butterfly Loop is easy to tie and very popular. Figure eight in the bight is often used by climbers apparently, but it seems to me it takes up more rope and is more prone to jamming. I have seen an argument by cavers against the use of the Alpine Butterfly Loop when tied using the "two twists" method, as incorrectly forming the loops can create the False Butterfly, which is hazardous. The danger of mistying it in a more subtle fashion using the "hybrid" method is discussed immediately above in this thread.
End Loops:
Figure 8 as an end loop is widely used in climbing, easy to form unless you are tying the "rethreaded" version, jam prone.
Alpine Butterfly Loop is liked as an end loop. David Delaney's web site advocates the Alpine Butterfly Bend Loop as slightly less jammable, which puts the working and standing parts in different parts of the loop than the standard Alpine Butterfly tied as an end loop.
The Zeppelin Loop is similar, perhaps marginally better. Both Alpine Butterfly Loops and the Zeppelin Loop require some practice to be able to tie quickly, but can generally be tied slowly if you forget how by starting with an overhand loop, staring at it, and remembering the structure of the knot.
There is, of course, the Bowline and its variants (Water, Double, etc.). Easy, compact, usually reliable. As an aside, I found that when practicing the "lightning method" of forming the bowline using a collapsing slip knot, you must take care to push the bight through the loop from the standing part, not the working part (or else you get an Eskimo, or Sideways, Bowline), and you must insert the working end through the bight in the proper directly (or else you end up with the Dutch Marine (Cowboy) Bowline). Both of these are described by various sources as being either less or more secure than the standard version. It is harder to get it wrong using the standard rabbit-hole-tree method.
In keeping with the subject of this thread, I'll count up the most "useful" of the knots above. I count as separate the "same" knot if they are tied using different methods, and technically different knots the same if they are or can be tied using substantially the same method.
1. Double Sheet Bend (an easy bend)
2. Carrick's Bend (near ideal bend)
3. Slipped Buntline Hitch (an easy hitch)
4. Zeppelin Bend using "b and q" method (easy to learn, near ideal bend)
5. Zeppellin Bend and Zeppelin Loop starting with overhand knot method (alternative technique necessary for the loop)
6. Alpine Butterfly Loop in the Bight, and Bend, tied round the hand (easy to learn, near ideal bight loop and bend)
7. Alpine Butterfly End Loop, or alternatively the Alpine Butterfly Bend Loop as described by David Delaney, both tied starting with overhand knot (near ideal end loop)
8. Sailor's Hitch, and Gripping Sailor's Hitch (good round pole hitch, near ideal lateral hitch)
9. Adjustable Grip Hitch (good adjustable hitch)
10. Figure 8 as stopper, bight loop, end loop, bend (versatile and easy)
For me, all of the above is rather theoretical since the only time I ever tie a knot in something that is not a practice rope is to attach a kite string or hitch my dog to a post. I don't even tie shoelaces any more. Please let me know what you think.