Basics
The 1-2
This is the golden rule of Minesweeper. If you can understand it, most of the other rules follow as a matter of course.
Suppose you've cleared some squares, and encounter a flat row of uncleared squares. Beside them you see the numbers 1 and 2:
There may be other numbers in the cleared squares as well, but we can tell a lot from just the 1 and the 2. What can we say about the two uncleared squares just below the two numbers? They can't both be mines, because that would overflow the 1. So at least one of the 2's adjacent mines must be behind the outer square:
The 2's other mine must be in one of the two uncleared squares under the pair of numbers. It doesn't matter which -- if it is behind either one, the 1 is satisfied, and so we can know the square on the outside of the 1 must be clearable. We are left with this (note the green means that square can be safely cleared):
Though we can build a lot of rules off of that simple rule, you should still find yourself using the 1-2 rule itself in its simplest form quite frequently.
The 1-1
The mine we've identified above adjacent to the 2 effectively reduces the 2 to a 1 that still must be resolved. So we effectively are left with two 1's adjacent to each other. In fact, any time you have two 1's adjacent to each other, hemmed in by a mine, a wall, or a cleared square, you can know the uncleared square outside the second 1 is clearable. Here's a 1-1 hemmed in by a wall, which, like before, gives you a square to clear outside the second 1:
Here's a 1-1 hemmed in by an already cleared square, again giving a new square to clear:
And here's an "effective" 1-1 -- the 3 is reduced to a 1 by the two adjacent bombs, with a clearable square in the same position:
The 1-1 appears frequently, and in fact gives you a little bit more. The above examples are all just working along a flat section. In truth, none of the squares adjacent to the 1 can be mines, if they haven't been cleared yet:
As is the case with all the rules listed on this site, reflections and rotations of this rule work just as well.
The 1-2-2-1
The 1-2-2-1 follows from applying the 1-2 rules twice in opposite directions. Here, the 1's get satified by the two mines placed under the 2's, so their other two adjacent squares are clearable:
The 1-2-1
The 1-2-1 also follows from applying the 1-2 rules twice in opposite directions, but overlapping a bit more. Additionally, you get to clear the middle square because the 2 ends up satisfied:
The 1-3-1-Corner
This rule essentially derives from two 1-2's at right angles to each other. You can infer a mine in the corner, and hence two clear spaces outside of the 1's: