Many smaller (and generally cheaper) machines use DC motors, mainly I think because they can easily be speed controlled so saving on mechanical speed reduction. (Mechanics more expensive than electronics these days.) Some DC speed controllers are a bit fragile, especially it seems those fitted to small mills and lathes (e.g. the X1 mill, see various tales of woe here). Brushed motors are traditional but the brushes can wear out. Brushless require electronics to drive them – they have only appeared on amateur machines in the past couple of years, so reliability perhaps an unknown quantity. They can also be more powerful in a given size I think. I have a small CNC mill (Denford Novamill) with a 1/2 hp brushed DC motor run from a "proper" KB Electronics controller and more than once I have stalled the drive (doing something silly) – once the fuse blew but the drive survived, so they can be robust.
Single phase induction motors are cheap to make, very reliable, quite efficient, but need AC to make them work – but AC comes out of the mains so isn't a problem. They can't be efficiently speed controlled so you need belts/gears. Because of the way they work their torque varies cyclically slightly out of phase with their rotation, hence vibrate a bit. But most small machine tools have done very well with them for decades. Also (in reasonable power versions) they need special means to make them start, usually consisting of an extra winding and a capacitor, and generally a centrifugal switch. The capacitor and/or switch sometimes give trouble. By changing the way the extra winding is connected they can be reversed. Maximum power is generally up to 3/4 hp or so. They generally come in 2-pole, ~2900 rpm; or 4-pole, ~1425 rpm (on 50 Hz). Poles and volts are unrelated.
3-phase induction motors are also cheap to make, very very reliable, efficient, but need a 3-phase supply. They start on their own and can be reversed by swapping over two connections. Again, 2-pole and 4-pole are common. Speeds (on 50 Hz) the same as single phase. Industrially they sometimes run on 415V phase-to-neutral, in a star winding configuration. Such a motor will need rewiring in a "delta" to run on a standard 3-phase "domestic" supply which is only 230V phase-to-neutral. This may be easy or hard depending on construction! Rotation is very smooth and vibration free (as long as the motor is properly balanced). The magnetic circuit of a 3-phase machine is more efficient so they can give more power in a given frame size.
These days a 3-phase motor can be fed from an "inverter" or Variable Frequency Drive (VFD). These allow you to vary the speed from virtually stopped up to rated speed up to (in principle) as high as you like. I run my mill up to 75 Hz, 50% over plate RPM. The VFD can be fed from single phase (usually), or the higher power ones 3-phase.
There is nowadays no point whatever in having a 3-phase supply connected to your house unless you need a lot more power than you can get from a standard single phase supply, as you can run a 3-phase motor from a VFD. Indeed it's better using a VFD because you get variable speed as well.
In summary,
- If you are looking at a hobby size machine, if it's a type that has a DC motor then ask around on the ME forums to see what sort of reliability people are getting.
- There are some machines that are fitted with a 3-ph induction motor and VFD – I think these are likely to be better and more reliable. Industrial grade VFDs are readily available and quite cheap if you needed to replace the original one.
- If you buy a used machine, you would probably be very happy with a standard single-phase induction motor but could improve it by fitting a 3-phase motor and VFD. (Or if you get a used industrial machine with 3-phase, add a VFD and use it on your single phase supply.)
- Don't worry about getting a 3-phase mains supply unless you want to run a big muffle furnace or electric kiln!