Over the years Remington has made a lot of different .22 rifles, from single-shots to semi-autos. In my personal opinion the 510-series rifles were some of the best. They are all bolt-actions, and range from a single-shot small game and plinking rifle to a full-blown match rifle, and some models had smoothbore variations for shot cartridges. Some of the repeaters had box magazines, the others tube like the 512, also known as the Sportsmaster:
They're full-size rifles(which made shooting my Dad's difficult when I was small), all steel and wood. I've had a chance to shoot several, and they all had good triggers; some were heavier than others, but all broke cleanly, an important thing for accuracy. Sights ranged from notch rear and bead front to match-type adjustable. And they were capable of fine accuracy.
I found this one some years back, and bought it mostly because it is the same model as Dad's. Like a lot of old rifles it had a problem, but it only cost about $15(shipping and all) for the parts, and it's worked well ever since. And I've been very damn glad I bought it.
When I say capable of fine accuracy, I mean it. A while back I decided to try some different ammo in mine, and with Federal Lightning(I've had such good results from the stuff it's become my default starting ammo for testing) it shot groups of 3/8" at 50 yards(I had a very good day), and would do it consistently. They vary, of course, depending on individual factors, but with the right ammo most of them seem capable of this level or close to it(yes, it was shot from a solid bench with sandbags; I don't think I could do that offhand with anything).
This one has the notch/bead sights, so about a year after I got it I mounted a scope(first time I'd drilled & tapped a receiver for a scope mount, and I sweated blood making sure it was square and that I didn't accidentally drill through into the chamber). The stock was a little beat up(surprise, surprise), so I scraped the old finish off, sanded lightly and gave it multiple coats of Parts availability is generally good; Remington made a lot of this series, and many of the parts interchange. Also, made as they were, it's pretty rare for something to break or wear out as long as they're treated decently.
All in all, if you run across one of these you should definately think about giving it a home; you could do a lot worse.
Additional notes: the main thing that does seem to happen to these is that the nose on the sear that acts as a bolt shop will get chipped or worn and will let the bolt slip out when you pull it back. The other thing that affects this is one of the screws on the left side of the receiver; if it's not staked tightly and gets a little loose, that can cause problems. Both are easily fixable/the sear takes a little more work/.
This was the rifle I was going to use for the .22 postal match. If (pick from the list) hadn't happened.