1. Lever.
2. Bolt, if a left hand action can be found.
3. Semi.
4. Falling block
5. Break action
6. Pump
7. Right hand bolt.
I should probably explain that. Levers are fast, well balanced, slim and friendly, at least to a cowboy shooter like me. They're just about perfect for woods and brush country, where ranges are short, shots are sudden, and you don't want slings, telescopes and bolt handles snagging on every passing branch. Their great limitation, of course, is that the traditional Winchester 94 or 86 doesn't handle pointy bullet, high intensity calibers well, which limits their utility for long range shooting, as is often found in the mountains or on the plains. I know there are Savage 99s, Winchester 1895s, and BLRs, which can be chambered for more modern calibers, and mount scopes more easily, but instead of solving the problem, they seem to me to combine the vices of both levers and bolts into a sort of misbegotten not fish, nor fowl, nor good red herring abomination in the eyes of the Lord. Your mileage may vary, of course.
Bolts pick up where levers leave off, at the edge of the woods. When chambered for high speed, flat shooting calibers, and equipped with telescopic sights, shooting slings, excellent triggers, and free floated barrels, they are much better adapted to the deliberate, precise shooting that's more to be found out in the open.
Semis gain you a number of things. They are fast to fire extra shots. They hold plenty of ammunition. A box mag is quicker and easier to reload than an internal mag or tube. However, they do these things at the cost of increased weight, bulk, and cost. Additionally, most of their virtues are irrelevant to hunting, so they're not really ideal as a field gun. I ranked them as high as I did because I'd surely choose one for fighting, but I tend to have more occasion for hunting than fighting rifles.
Single shots tend to be specialist guns. They can be made very light and compact in comparison to a repeater, and the falling blocks can be very accurate as well. In most situations, though, a repeater is cheap insurance. One caveat to singles, is that the ease of chambering them for any cartridge whatever, regardless of length and other factors that restrict repeating rifles, is that people tend to do silly things with them. A single shot carbine may make an excellent deer rifle. Make sure of your shot, and you're good. Less so for elk. They frequently take several shots, and why complicate your life unnecessarily? But look at the Ruger No. 1 Tropical. A single shot dangerous game rifle? Surely the market for these is someone who wants to play with an African caliber at the range, and doesn't want the expense of a true safari rifle. But if he actually uses it for what it claims to be for? Well, think of it as evolution in action.
Pumps exist. They seem to work. Some people like them. Enough said.
Right hand bolts. I can shoot right handed, but as it involves fighting both eye dominance and the absence of a significant portion of that trigger finger, I get a more harmonious outcome from the other shoulder. Shooting a bolt wrong handed is just annoying, so I don't, if there's a choice.