Most people use condenser microphones. One of the most popular is the Audio Technica AT-2020. It's running for around £80. There is a USB version for £150, but I think it would be cheaper to go for the regular one.
The only condition with that is that you need a mixer, or some sort of power source for it. Condenser microphones operate at slightly higher voltages, so you need something to power it.
A very simple example of this is the Behringer Xenyx 302 mixer. Very simply, you plug the USB into your computer (to power the device, the microphone and to transfer audio) and you have a couple of dials to control the volume as well. The 302 is running for around £37 on Amazon, so this is a total of £115 (cheaper than the USB version by itself).
In addition, it's worth getting some sort of stand, no matter which microphone you go for. You can get a simple one for a tenner or so. You will also need an XLR cable for about £4 or so. That would be your absolute basic setup. There is room to expand though...
It is also advantageous to get a shock mount. The Samson SP-01 works with the AT-2020 microphone. This connects to the microphone stand (watch out, it's a bit heavy at this point) and prevents vibration from causing audio disturbances. That is roughly another £32 or so (£151 grand total).
If you intend to record audio, it may be worth getting a pop shield for £10-£20 (check the reviews) or perhaps making your own. A pop shield gets rid of harsh "P" sounds and the like from your audio, so you can sing along and have a cleaner recording. So £170 all in for the "complete" range of equipment.
Regarding recording, I think the Behringer 302 mixer has software (well, you download it) to handle recording and editing. A free alternative is something like Audacity, just to get the levels right and to remove any noise from the recording and generally clean things up.
I have used and tested all of those products, hence my hefty review and promotion for them. But when I looked, that really was the cheapest option. There are other USB condensers such as the Behringer C-1 or Samson CO1U. There are also Blue Microphones, although I have heard mixed reviews about those. Your best bet is to look them up on Youtube and listen to people playing those sorts of instruments through them. The stand, filter and shock mounts are separate to the actual microphone you get, but it can be well worth investing in in the future.