Some are solderable and some are not (look in left column). Those that aren't solderable can be removed by much higher heat -- often done with a salt bath containing melted KOH or NaOH and a little NaNO3 -- or by mechanical means. I am not aware of any simple solvent that works at room temperature to remove them.
For some sizes, a brief pass through a flame will take off the enamel. Too fine a wire will also burn.
I usually use an Xacto knife blade and pull the wire over it to scrape off the enamel. Steel wool can be used, as can fine sandpaper (400 - 600 grit wet or dry used wet works pretty well).
If it's the enamel on the magnet wire, you can use brake flute to remove it. This is what i use and it work.
Just take care for it doesn't get onto your clothes.
Mitch, there are some very thin enamelled wires made specifically for wiring prototype circuits. I got a big spool of it on eBay last year so it's available. Works pretty good too as you can easily daisy chain connections. It really needs the 'pen' tool though to dispense and wrap the wire properly. That Ive not been able to find, but someone told me there was a company in your neck of the woods selling this stuff.
For enamel/magnet wire, I just use a green Scotch Brite pad and a swipe ro two.
The rest of my breadboard wiring comes from Cat5, stripped to the right lengths.
Wire Wrap stuff has teflon insulation, and most wire wrap tools come with a nifty stipper. It's a fast and easy way to do a prototype without solder or a breadboard (logic stuff).
If you want to keep it, it's also good, as the bond of the copper around the square pins doesn't allow oxygen in. IIRC, the space shuttle has wire wrap connections to the magnetic core memory.