Q:

What frequent patterns can be used to overcome Minesweeper obstacles?

Minesweeper, to those who do not know the strategies involved, is a difficult game.
What patterns exist in the game which I can take advantage of? 

Meta-context: this question is an offshoot of the current original question, for which the scope has been changed to be simply pattern-based. Feedback indicated this was somewhat too broad. This question has now been edited to address these concerns. Please contribute to discussion on Meta if you feel inclined.

A:

Small notes

Remember that the number of remaining mines is the clue number minus the number of surrounding mines. This is important in patterns like 1-2-1; 3-2-3, for instance, also works if there are two mines next to each 3, since they both need 1 mine to be solved.
A one by itself on a corner always indicates that there is a mine
A two by itself next to a convex corner always indicates two mines
Patterns that work on horizontal walls obviously work on vertical walls, and vice versa.
Patterns do not work when there is interference from adjacent clues or unknowns; i.e. when a pattern is in a corner or has another unknown near it.

The double-click trick
On the Windows version of Minesweeper, if you click on a clue with both your left and right mouse buttons, if all that clue's mines have been identified, then the game will remove all the other surrounding unknowns. You should always use this to clear spaces, unless you are guessing.
Additionally, you can use this to check visually if all the spaces surrounding a clue are mines. If you need two mines, and two squares are highlighted when you use this trick, you know they're both mines.
You will primarily use this trick to identify patterns quickly through visual recognition of shapes.
Note: if you have a mine incorrectly marked, this will cause you to lose the game - but then again, you would have anyway.
The 1-2 pattern
Whenever you see a 1 next to a 2 against a wall, you know that the space opposite the 2 is a mine. 

 

This is because there can only be one mine in the two spaces adjacent to the 1, which means the second mine must be adjacent to the 2 in the final space.
Note that in the second case, since there is a mine next to the 3 and 2, they need 2 and 1 mines respectively, matching the 1-2 pattern.
The 1-2-1 pattern
One of the frequently-encountered patterns along walls in Minesweeper is the 1-2-1 pattern. This is an extension of the 1-2 pattern, in that two of the pattern exist side-by-side. In this case, as an extension of 1-2, the mines must be underneath the ones.

  

Note again that in the second case, the 2-3-2 are all adjacent to a known mine, which means they need 1-2-1. In the third case, we know that there is a mine next to the two by the one and corner in olive-green. therefore, the 1-2-2 matches the 1-2-1 pattern, implying there are two mines in front of it.
The first position was honestly just dumb luck. Yes, the 1-2-1 pattern does chain.
Also note that it is very rare to actually encounter a 1-2-1 specifically. Usually, these are by reduction from surrounding mines.
The 1-2-2-1 pattern
Whenever you find a 1-2-2-1 pattern against a wall, you know that the two squares under the 2s are mines. 

Unfortunately, I found myself unable to generate a true 1-2-2-1 case in a reasonable amount of time, but here is a valid position anyway that demonstrates the point.
The 1-1-X pattern
When a 1-1-X are sequential against a wall and against the edge of the grid, the space under the third clue, marked here with a question mark:

This is because the first two spaces must contain a mine by the first clue, which means both the first and second ones are already guaranteed to be satisfied. If there were a mine in the third space, it would conflict with the second one.
This pattern is, admittedly, a lot less useful. However, it has the possibility to save your game, especially when you're stuck.

