The ends of shoelaces are called aglets. The word comes from the Middle French aguillette, which means needle, and the Late Latin acicula, which means ornamental pin.

• Nail varnish: merely makes the frayed bits more solid, although this can help when attempting to force them through lace holes.
• Grabbing one of the frayed bits, sticking it through the hole and pulling: unless you have exactly the right kind of shoelaces, this merely unravels them further.
• Sucking them: aside from hygiene concerns, which I gave up on in desperation, this has the same effect as nail varnish except even less useful as the stuff doesn't harden.
• Cutting off the frayed bits: the ends, surprisingly, come apart again in a matter of seconds.

In conclusion: shoelaces cost 50p. Buy some new ones.

The frayed ends of shoelaces can have superglue (crazy glue, cyanoacrylate) applied to them and then twisted into a sharp point. Then when they harden you'll have a shoelace-end even better and more hardwearing than the original.