Carbines
Short, handy carbines, like Marlin and Winchester Model 94 levers, .30-30's, the Ruger Mini-30, etc. should have a small, handy low power scope with a wide field-of-view for quick target acquisition. Large, high-power variable scopes are the worst choice. They have a short and critical eye-relief: you have to hold your eye quite close to it, and at a precise distance, too, detracting from the quick-handling nature of the gun. Besides being slow to acquire a sight picture, they are heavy. A large objective (front lens) lessens the handiness in heavy brush, and must be shielded from knocks at all times. High power also means a narrow field-of-view, which means you can see large game close by, put your eye to the scope, and then spend a couple of minutes examining blades of grass and tiny insects trying to find it. Finally, carbines are of limited long range potential, so there is nothing to be gained by choosing a powerful, delicate scope.
Good choices for carbines are usually fixed power, low-power scopes. They have the widest field of view, good for quick target acquisition. The fixed power means less moving parts in the mechanism, making for a sturdier instrument. Being low power, they do not need large front lenses to gather light well in dark wooded areas. Two to Four power scopes are the most popular in this class.
Shotguns
What holds for carbines is also true for slug-firing shotguns. With long receivers and even longer recoil, shotguns often call for Intermediate Eye Relief scopes, to prevent eye injury from the recoil. Scopes in this class typically have a magnification power in the 1.5x to 3x range. Red Dots are another popular choice (though shootings slugs can make short work of cheap Red Dots).