I'm the exactly opposite of the OP. Currently, I'm dealing with the first double sink I've ever had, and I don't like it. Difficult to wash cookie sheets and the big Dutch oven. And two drains to have to keep clear, instead of one.
Mostly, in a single sink, I've used a dish pan to hold the hot, soapy water. I wash a few dishes and put them in the empty part of the sink, and rinse them when there's a small stack. Then repeat until they are all done.
Some people leave the water running and rinse each dish/cup/fork as it is washed. Some people put everything in the drying rack and pour or spray water over them when everything is washed.
Some people just plop the dirty dishes in the sink, put the soap on the dish cloth, run the water and wash and rinse each piece.
I've only once seen a single sink that couldn't hold a dish pan. That's in my brother's house, which has a very badly remodeled kitchen. He has a small, but very deep single sink--it's about the size of one side of a double sink. He puts everything possible into the dishwasher, and washes everything else by running hot water and washing and rinsing each item in a single step.