I want to build simplest interferometer which should be able to measure movements down to fraction of wavelength.
What is the simplest scheme for that, and what are the requirements for a laser?
I have a bunch of laser diode-based ones, and I guess they might be not coherent enough... Are green DPSS ones any better?
Well, one problem you are likely to encounter is that your setup will likely be vibrating with amplitude on this order.. do you have a floating optical table?
Coherence is probably not that important. At a minimum you will need a beamsplitter, two mirrors and a diode or some other way to measure the interference pattern. A lens or two to magnify the pattern will also be helpful.
Lasers aside, the mirror translation stages are likely to be your biggest hassle. If you want to measure sub-optical-wavelength distances, then you should expect to need translation stages that are stable to this sub-micron range or your measurement will wash out. That is, you can have static interference patterns, but if you expect to move mirrors and count fringes to do a length measurement, then your mirror-mover should be stably controllable to the length scales you plan to measure.
On a practical level, this means you will really need a quite solid optical surface like, say, this one, and your mirrors should also be stably mounted with precision screws. If you have a moderate budget then this is nothing to balk at and I expect$^1$ you should really be able to build a quite reasonable one for something like $$100 or $200.
This is not to say, however, that you can't have something really good for really cheap - it just says that you won't be able to finely control the interference pattern. If seeing the pattern is all you really need, then resources like this instructable seem to show that getting a Michelson-Morley interference pattern is relatively easy if you have the patience.
$^1$Note, though, that I'm a theorist.
You should also consider the laser's frequency stability. My understanding is that the frequency of light can go up and down by some nanometers as temperature (and perhaps some of the electrical inputs) of a laser diode varies. Searching seems to turn up quite a few articles on stabilizing their output, exactly for this purpose.
I'd suggest a Mach-Zehnder interferometer configuration because it may be easier to align.
Before you go out and spend all your money on developing an ultra-stable source, why don't you try a simple, free running He-Ne laser. I am suggesting a He-Ne laser because the output mode is very clean and this is important in practical interferometry.
If frequency of the laser is a problem, you need to figure out over what timescale are your measurements being made? Typically, gas lasers are stabilized externally by locking their frequency to a cavity (the Pound-Drever-Hall technique is well known).
If you are using a diode laser, things get a bit more complicated. You will first need an external cavity configuration (check out Using diode lasers for atomic physics by Wiemann, Hollberg et al.,) along with precision current and temperature control. Typically free running diode laser linewidths are in the order of 40MHz and with grating feedback, you can narrow the linewidth to about 1MHz. But without thermal control, the wavelength can drift by several nm. If you really want kHz linewidth and long term stability you need lock your laser to an atomic line and use active stabilization.
The point I am trying to make is that, the more demands you place on the source the higher the complexity of the setup. So, I'd suggest trying with the simplest available stuff, hell even a laser pointer to see what you can do with it.
