Which conformal maps should one have memorized? I know this is a bit vague and there's no end to the conformal maps, but I'm just curious which ones you've memorized out of sheer usefulness, particularly if you work in an area related to complex analysis. Which ones should one be able to derive as well?
 A: There aren't that many to memorize, and you should be able to derive them all. Know how to use linear fractional transformations, including regular linear transformations like rotations and translations. Know what the exponential and logarithm maps do. Know what taking powers and roots looks like. You should be especially familiar with how to map the unit disk to the upper half plane and back. 
You should also know the Schwarz-Cristoffel and Riemann Mapping theorems.
The best way to get comfortable with conformal mapping is simply to do a lot of problems. I know Churchill's book has many problems on this topic, and so should any other standard text. There are a very small number of speciality tricks (e.g. mapping an ellipse to a disk) and you just pick those up from seeing them used. The vast majority of conformal mapping you do (I'm assuming this is for coursework) is going to be finding clever ways to compose the basic maps I listed above. This is a skill you pick up through practice.
