Internal Singer motor wiring

The machine was a Singer 201K3 but all Singers can have the same problem as this one.
When you come across an old Singer motor (1950s or older), it should be inspected thoroughly. If you have no idea what you're doing with electrical things, motors powered by mains voltage aren't a good way to start learning. Don't just plug it in and turn it on. While I'm on the subject, if the motor is hard wired (like the early ones), don't just switch it off, unplug it too. You'd be shocked to learn that the electrician crossed the neutral and live wires on that power point (pun intended).
Being someone who has quite a lot of experience with electrics, I removed the carbon brushes - remove them and put them somewhere so you can put them back in the exact same side and position as you removed them. If you put them back differently in any way, they won't make contact as well as before and the motor will be weak for a while. If they're short, you should replace them. They're only a few dollars and if you wear them out the springs will damage the commutator.
I then took the motor apart and the stator (the bit that doesn't move, also called the windings) wiring was as per this photo.

The wires (black and red)

In case it's not obvious, these wires are not fully insulated. I touched them a few times and the remaining insulation fell off. So where to now?
If you're not also extremely competent, confident and comfortable with electrical work, you have these options:

1.  Take it to a small appliances shop and have it repaired.
2. Have the motor reconditioned (re-wound) by a motor repair service (they do exist).
3. Swap the motor with a known good one (or a new one, but be careful, new Chinese made motors are often as dangerous as the one above).
4. Take the motor and light off, replace the hand wheel and pop it into a treadle cabinet.

Assuming you're all good to continue, you need to now make these two wires safe.
You will need a soldering iron, solder and some heat shrink tube insulation. If you don't have the latter, get some or you won't be able to complete this. Do not use electrical tape. It will melt off as soon as the motor gets warm, as will anything else. The heat shrink tubing needs a soldering iron's heat to shrink and will only melt if you burned it in a fire.
The rest of the wiring in the stator is held in place and not insulated with rubber, so we don't need to worry about it.
At the brush mounts, the wires are soldered to a copper ring, which is in turn soldered to the square brush contacts, which you can see in the picture is on either side of the hole at the end.
I was not able to remove the copper ring because of the amount of solder, so I cut it as close as possible.
Remove any remaining ' insulation'.
Now, cut a piece of the heat shrink insulation that will fit as snugly as possible over the bare wire and shrink on some new stuff (use a less hot part of your soldering iron and gently stroke it on all sides until it's hugging the bare wire).
Now solder it back onto where you cut it.
Do as Singer originally did and smother the whole end of the wire with solder. This is difficult because you need more hands than you were born with but you need to make sure these wires will never come loose.
It's unlikely they'll ever be pulled because only the rotor (armature) moves and you're going to ensure that it doesn't move anywhere near these two wires, but make sure it doesn't come off as you reassemble it.

Afterwards. Apologies for the fuzzy photo

I wasn't entirely happy about that millimetre of red wire that is exposed but there's absolutely no chance of it ever touching any conductor.
Repeat with the other wire then clean the commutator (the copper strips that come in contact with the carbon brushes).
Now reassemble the motor body, while watching the wires carefully as you put the end on. The wires will move differently now that you've changed the insulation. Make sure that the whole soldered area easily clears both the rotor's wiring coils and its contacts (the copper strips). I stress easily because the wires could settle slightly and move a little more when the motor warms up. Just imagine all that could go wrong and check that you've made sure it can't happen. It doesn't matter that it's a thousand times safer than before if it electrocutes someone because you didn't put enough solder on, left too much wire bare or used electrical tape. Do everything and your conscience will be happy.
So it was much more work than I expected, but I was extremely happy with the result. This motor should last many decades without anything more than the occasional set of carbon brushes.