What advanced cycling skills are useful for a commuter or city rider?
Some important skills for commuters:

Looking directly behind you without turning.

This is a surprisingly difficult skill to master. When a rider looks directly backwards, it is common to turn in the direction they twisted their head. It takes a lot of practice to make resisting that turning automatic. The importance of looking behind you in traffic should be obvious.

Rapid countersteering

It is not uncommon to come up to an intersection when a vehicle in the crossing direction will come in front of you, and you will not have enough time or space to stop safely. Sometimes what is required is a countersteered* turn into the direction the crossing vehicle is traveling. This requires a quick turn of the handlebars in the opposite direction you want to turn, which will cause your center of gravity to rapidly swing to the side you want to be going. Returning the handlebars to turn in the direction you want to go completes the turn, and by doing so you can turn very rapidly with a very tight radius. 

You might not be able to stop, but it can be preferable to make a sudden turn rather than crashing into something.

Stopping suddenly

Not many riders are highly skilled in using their brakes. Practice slamming on your brakes, especially using only the front, and get a feel for how much force it takes to get your back wheel off the ground. Practice to the point that when you are in a panic situation you don't simply clutch the levels with every ounce of strength, practice so modulating your brake power is a purely mechanical, automatic skill. Also practice stopping on sandy, wet, and icy surfaces.

Lesser but still important skills


Watching for people in parked cars that might open their door right in front of you while biking alongside them.
Recognizing when one of the common collision situations are upon you and how to respond defensively.
Riding on slippery surfaces.
Never stop at the stop line where 18 wheelers are turning, always stop a good distance behind so the truck has room to turn.