Pure Combinations
Some beautiful rules can be applied every time you see certain combinations of numbers, regardless of what squares have already been cleared around them.
The 1-4, 2-5, and 3-6
These interesting cases will always tell you where three mines and three safe squares can be located. The 1-4 is most common; the 2-5 less so, and the 3-6 even less so (there's even a similar 4-7, so rare it's not even worth showing):
The 1-3-2
This one I stumbled across after years of just playing right through it. I ran a computer verification for it and it turned out even better than I had realised. It doesn't matter what you already have cleared around it -- wherever you see a 1-3-2 combination, you are guaranteed eight clear squares, and two mines flanking the 2.
The 1-2-1
Though usually along a wall (see the 1-2-1 in the Basics section), this rule can be generalised -- like the 1-3-2, wherever you see this combination, all these eight squares can be cleared: