Soldering thin polyurethane covered wire
I would NOT use a knife or sharp edged tool of any kind on thin wire.  Very fine sandpaper if you can strip a large area and then clip the end of the wire to just the length of stripped wire that you need to solder.

If you can only strip a very short area for soldering, then I would burn the insulation off with a soldering iron, and tin the stripped end and examine it before actually soldering the connection.

Using a knife can cause nicks in the wire, which will later break when things vibrate.

I used to use this trick when soldering wire-wrap wire.  I always burned the insulation off the wire, one of my coworkers preffered to use a wire stripper.  When we got stuff back that wasn't working, it was always the ones my coworker did - the wires broke off where he had stripped them.

You'll need to get a blob of solder on the tip of your iron, then poke the wire in to the blob so that the point of the wire touches the surface of the soldering iron tip.  The point of the wire to the soldering iron tip helps to heat the wire itself and melt the insulation from the inside.  The blob of solder helps to heat the insulation from the outside.  It takes practice to be able to do this quickly and accurately, but at least you won't have to worry about the wire breaking later.