This post won't cover the other mechanical pump types. Diaphragm and piston oilless pumps for backing turbos will be handled elsewhere, and the many other kinds of mech pumps in use (scroll, roots, and so on) are mainly used in very large systems which I doubt would be of much interest here.
Lutz, since you've probably rebuilt more of these than I, how about chiming in with some more specifics in a reply to this?
Vane pumps are ubiquitous in the surplus markets, and still available new at medium high costs. Since they are pretty easy to determine if they work at all, you can often find them on ebay needing only a little further work, though the "little" may mean what amounts to the same effort as a rebuild of a small engine. What you will want for any real vacuum system will be a two stage one, the single stage ones for air conditioning maintenance just aren't good enough for anything but that use -- they are more or less more convienient alternatives to the old re-purposed refrigierator compressor, with the lubrication taken care of a little better. So if you're going to start building a system, a decent two stage mech pump is the main thing you're going to want to have first to get "feet wet". I believe that everyone but the vendors prefers the slower speed versions, belt driven off the usual induction motor, as they last a lot longer and are quieter, and more often than not, draw less power too. The faster running direct coupled ones can run bigger clearances due to oil-seal inertia, and run thinner oil as a result, so their advantages are cheaper to make (but usually not to buy) and can maybe run the same oil as a diffusion pump to avoid any chance of oil cross contamination, if that oil is Lesker's Diffoil 20. The "good" diff pump oils don't lubricate well enough to use in a vane pump in general, and the slower mech pumps need thicker oil.
When you find one that is either seized or just doesn't seem to work right, 99% of the time the trouble is going to be rust based. You'll either have leaks due to rust pits, or even more often, a vane that is stuck in the rotor and cannot slide in and out to match the sort of Wankel engine shape of the chamber the rotor moves in. The fix for this is fairly obvious -- you take the thing apart, and clean out the rust, put it back and try it again. These guys seem to be amazingly resistant to small rust pits, and you can run them back in for a few hours on kerosene to kind of re-lap the vane end seals to very good effect -- in this case, leave the rust in there -- it is a great abrasive, and it will wear everything to match again nicely. Then flush a couple of times with something cheap (more kerosene or something similar), and then put in the real oil (which isn't cheap if it's good stuff). Do these run-ins with the intake blocked, and let the pump get good and hot while doing it. I do this outdoors so I don't have to breathe the fumes. After putting in the first good oil, performance won't get to be good till the last bit of kerosene has been allowed to evaporate -- some more hours of blocked off runtime will usually do this, or better, have a tiny leak so some gas is going through there to help carry the kerosene vapors off. Mech pumps don't enjoy an unblocked intake at STP -- they will tend to spew oil in that condition. In a real system, this is usually only a very momentary condition, and livable. But don't run them long times with unblocked intake -- it can be a large mess to clean up, and if you've got the good oil in there, an expensive one.
Why do all these guys show up with rust? Well, at least two reasons. Any big hunk of metal will cool at night, then stay cool while warmer air comes along and presents water condensation. That's not a major one unless the pump is sitting for a long while. In use, there is always water vapor in whatever you are pumping. When this is compressed in the pump, it becomes more than 100% relative humidity and tends to condense in the pump. This is such a prominent and bad effect that most 2 stage pumps have a deliberate air bleed intake between the stages, as so much can build up in there that the pump, which is a positive displacement device, can hydro-lock on it and refuse to spin at all due to that. A key clue this is happening, or about to, is the sound of the pump changing and sounding kind of "farty". Go open that bleed right now. I've not noticed a big difference in pump performance with it open, so I usually leave it that way here.
Mech pumps have a nameplate rating on how low they will go. It's a lie. That rating is for a brand new pump, baked out, with the intake blocked, run for hours, often with a trap between it and the gage. You will NEVER see that in real use. Or anyhting particularly close to that number when it is hooked to a real system that is outgassing or has even the tiniest leak. Count on about a factor of 10 worse than the nameplate and you'll do fine.
One reason for this is that even with two stages, there's a net compression ratio involved. If you go to the manufacturers spec sheet, you will notice that as the pressure goes lower, so does the pumping speed in liters/minute (or in whatever other units). And you'll find that that last factor of ten in pressure only happens when it's basically pumping at near zero speed, no matter the liters/minute rating at STP. We have found no great advantage here, when working on the slopy part of that curve, in going from a 1/3 Hp to a 1.5Hp pump, for what it's worth. Sadly, the small, lower power ones seem relatively rare in the surplus world, perhaps because anyone who has one isn't letting go of that gem.
Because everything has a finite vapor pressure, and vacuum oil is a thing -- it has vapor pressure. This means some oil vapor will come back out of the intake side of the pump, potentially contaminating things upstream with it. Various things are used to attempt to limit this, from micromaze traps to what amounts to a big jug with steel (or stainless) wool in it to catch and condense this so it can drip back into the pump. Cold traps are sometimes used, but they really don't do much for you for oil until they are at dry ice temperatures, and still don't do much for water till you hit liquid nitrogen temps -- water vapor pressure stays amazingly high down to pretty low temperatures (which is why freeze drying works at all). And they are only a temporary solution -- they need to be cleaned out fairly often, so we don't use them here. We use the jug with the stainless steel wool in it ourselves, as the best compromise for us, and in our case, since our diffusion pumped systems don't need full available performance from the diff pump, we use Diffoil-20 in both pumps, so no cross contamination is possible anyway. In this case, the big jug mainly acts as a reservoir to handle peak pumping loads more smoothly.
A real problem with mech pumps, even ones that claim otherwise, is what happens when you turn one off with a decent vacuum still on the intake side. They tend to allow oil from the lube resevoir to "suck back" into the vacuum system, which can mean a lot of cleanup work -- and remember, good oil isn't cheap either. Yes, many claim to have solved that one with some trick valve in the intake line. Not perfect, and one incident of this will make a true believer out of anybody after they have to clean up after it. The obvious solution is "don't do that". But that's not always practical. In our designs, the big jug, plus a fat up sloping pipe to the rest of the system take care of this fine -- the oil just drains back down into the pump before the next run. If you try to get too tricky, watch out for this one, though.
More times than not, taking a pump apart to fix a rust issue means needing a case gasket again -- the stuff they use tends to tear during takedown. I've had great luck with making my own out of similar thickness rubber sheet (buna) from hardware store sources -- the stuff they put behind shower stalls. If that's too thick, and sometimes it is, try something thinner instead -- find a knowledgable guy at the auto-parts store for example -- you'll have better luck with a local one than a big chain store in that I'd guess. I've also had fair luck using silicone gasket maker (from the auto parts store, not the caulk at the hardware store). The reason this dimension is sometimes critical is that the rotor end seal is simply a tight fit between the rotor end and the case. Most avaialbe gasket materials will work fine for this, as even the best vacuum these pumps can make isn't all that great -- it's more important to just not leak or dissolve in the oil than anything else.
This is why even though metal touches in there, we call all these "oil sealed" pumps. It's really the oil making the final seal, and that's why the correct oil viscosity for your pump is key to it working well.
So far, I've not had any trouble with shaft seals and never replaced one yet -- Lutz may have more to say on that one, however. I have had to fix or re-make shaft couplers on direct drive pumps, however, and since I have a machine shop I just make another one. I guess most would have to go back to the vendor at that point and order one.
One nice thing about these (which really don't have much nice about them if you have other options) is that when running with a decent vacuum into them, they don't use much mechanical power, though the usual induction motor will still draw a lot. I'd suggest that anyone who is sensitive to that either buy a special high efficiency induction motor, or go to a DC treadmill motor for their setup (but get spare brushes if you do that). It will save power and heat in the long run, which for some is a nice payoff -- me since I'm on solar photovoltaic panels here, so saving a few hundred watts is a pretty big deal. At roughly 600 bucks per 200w of panels (when in full sun, only), well, you can work that one out.