A couple of years ago I had a go at propagating mint plants from various sections of stem and/or root, and it seemed to go quite well.
Recently, I spotted a Twitter post (sorry, I forgot to bookmark it for reference) that suggested it would be easy to take softwood cuttings and root them in water, so I decided to give that a go as well.
As you can see, it seems to have worked rather well:

One the left we have two cuttings of black peppermint, and on the right, two of apple mint, all trimmed to just below a pair of leaf nodes and then left to stand in a small vase of plain tap water.
The black peppermint was especially keen on the water-rooting method, producing new roots within a couple of days. The apple mint took a little longer; it was four or five days before new roots showed and as you can see they’re not quite as well-developed as a result.
Good enough for potting-up though. All four plantlets have now been transferred to small pots of peat-free compost and I’ve started another four cuttings off: two more black peppermint, and two swiss mint.
Peppermint tea, anyone?