So how should I go about cleaning a soldering tip sponge?
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1\$\begingroup\$ My first thought: "carefully". I always manage to get pointy bits of solder stuck in my thumb when I clean the sponge. \$\endgroup\$ – Simon Richter Aug 13 '14 at 13:06
I usually just shake out the bits of solder, and toss it when it gets too worn out. Considering that you can get suitable soldering sponges in 6-packs at the dollar store (and can cut them into quarters if you want a pack to last longer), this is not usually a huge deal.
I use brass wool. Sponges are not used as often as they used to be.
I use brass wool like this: http://www.gotopac.com/599B_02_Tip_Cleaner_p/599b-02-hak.htm. Though you can just use the wool alone, you don't necessarily need the holder for it. It's a bit more expensive and replacement wool is under $5 and it tends to last longer than a sponge.
I usually just soap it up good with something like dawn dish soap, work it around, and rinse it out.