I've recently dug my quarry/mine down to a certain level where it has encountered a large mineshaft section and I am building a mine cart track through it to a nearby ravine and caves (because, hey, pre-made tunnels!) I've blocked off sections with fences so I don't get lost but there's still an annoying amount of spiders webs about. If I'm in the mine cart it would be fine but walking about and even nudging them slows me down.
I've tried axes, spades, pickaxes and even digging out holes above and pouring lava on them (which works, although has the significant self-immolation hazard, when I've been mining and am carrying a lot of coal/cobblestone/some diamonds I'd rather avoid).
I'm playing on peaceful (Vanilla MC, no mods) so I don't have to worry about the spiders themselves.
Best Answer
Hit them with a sword and you'll be rid of them in no time!
It is also possible to use shears to cut through the web.