The very best ring binders, as far as I'm concerned, have more rather than fewer rings. I did some experimenting for a year or so with every type of ring binder I could get my hands. My intention was to put a few of the countries I collect in ring binders so the pages would lay flatter than in a two-post or springback binder, the other two main types of binders.
I looked carefully at every type of binder I could find at office supply stores first. All of those were very disappointing for a number of reasons. One, they're virtually all 8.5 x 11 inches which is an underwhelming size for a real stamp album. I realize many collectors, especially those on a budget, use these smaller school-type binders to hold their pages, but they look less like stamp albums than the school binders they really are. A constant problem with nearly all 3-ring binders is that they tear the pages at the rings. Most ring binders have a rough spot on each ring, and as you turn the page it rubs against the hole in the paper. Not good. And the fewer the number of rings, the more wear you get on each page hole. More rings wear paper less.
Another problem with 3-ring binders is that the pages flop around especially when the binder is standing upright. Three rings don't do a very good job holding the pages like a multiple-ring binder does.
The only 3-ring binders I've found that look like an album and are really well made are the smaller of the two Scott 3-ring binders. They're pretty decent. The larger Scott 3-ring binder is so enormous it looks to me like something you'd use for invoices at a trucking company. It's not pretty, and those huge rings will seriously damage your pages over time. And the larger the rings, the more distance there is between pages when the binder is open. With the smaller ring binder, you don't get that kind of industrial look
Both sizes sell for the same price, so you can bet that most people will buy the larger binder, thinking they get more for their money. Not very wise since the larger binder is heavier, less attractive, and holds too many pages (in my opinion) to be used comfortably. Plus more hole damage on the pages. But people always have eyes bigger than their stomachs. Both binders are deeper than normal albums, so they stick out of the shelf a few inches with the larger of the two looking truly massive. With one hand, I can barely pick up a full Scott 3-ring binder in the larger size. It reminds me of the bizarrely huge "Jumbo" size Scott International binder. Fill that thing up and good luck even picking it up.
Multiple ring binders work even better. Lighthouse makes very good ring binders, but they aren't cheap. More importantly, the pages you put in them are very expensive. LH pages are among the most expensive of all album pages, and that includes their blank pages, as well. They are very good looking pages, but I avoid them because they are enormously expensive compared to other pages.
Schaubek sells a 6-ring binder that's pretty decent. The advantage is that Schaubek sells pages that are just as good as the pages sold by Lighthouse, but they are much cheaper to buy. I'm not a big fan of the 6-ring look, myself, and they are pretty large to handle. I own a few but don't find them very appealing.
There are also ring binders from Lindner (but I hate their pages), as well as from Safe, and maybe others. None are cheap. All are well made. Davo does not sell ring binders, only 2-post binders.
in Europe, 4-ring binders are common. They hold the pages straight much better, in my opinion, than 3-ring binders. Various sellers have them. Some come with slipcases which is nice. You will need to buy 4-hole paper or purchase a 4-hole punch to punch holes in your own paper. They can take either U.S. size paper or the slightly taller European A4 paper common there.
My favorite ring binders are the 22-ring binders popular in Europe. You can hardly find them in the U.S. They have the advantage that the rings hold the pages straighter (less movement up and down) than 3-ring binders do. They look good as stamp albums partly because you don't "see" the rings as much as you do in a binder with fewer rings for some reason. All the ones I know have smaller rings so there's no giant ring phenomenon like with the large Scott 3-ring thing. All the ones I've used are well made.
Buying blank paper punched for 22 holes is easy in Europe, but not in the U.S. You have to have binders and paper shipped to you. I've done that many times. It's easy, but shipping costs aren't exactly cheap. Add 30-50% more for shipping, but you'll likely only order a few times, so not so awful. My two sources are Prinz Publications and Dauwalder's Stamp Shop, both in Britain. Each has a good website.
Prinze makes the Duchy 22-ring binder. It's a good binder, but the only slipcase available that fits is a heavy beige cardboard one they sell. It looks okay, and a bit retro. Binder alone is maybe $20. Cheap for a good binder which looks good.
Dauwalder's sells their own Salisbury binders with 22-rings and a mathching slipcase. They're not very expensive, maybe $35 with slipcase They're very well made and pages look great in this binder. It's one of the best.
If you buy their blank paper, you're all set. But you can also search around and try to find a 22-hole punch and punch your own paper. They don't come up for sale often, though. But I've managed to find some on eBay
from time to time. Punching U.S. card stock paper to make album pages is about the cheapest way to go.
Other manufacturers include Marini in Italy (hard to find) which also makes 22-ring binders and Yvert et Tellier in France which sells the same as well as 2-post binders.
If you must use a U.S. sized 3-ring binder, consider the smaller Scott binder. Or look at the slightly smaller 3-ring binder Amos/Scott sells on their website. It even comes with a slipcase. It's one of the best normal sized 3-ring binders available. I use some for storing covers in Vario pages. They're very good even if I prefer larger binders for actual stamp albums.
There may be others. I'd get a stamp album-sized (larger) binder and build my collection from there. Look at 22 ring binders for the best ring binders. But the others will still work.