In Sweden... You have a sink split in two sections, one that you filled with dishes+water, one for rinsing. My parents yhsed a dishwasher, and I later had a dishwasher as well.
In the UK, my sister uses a wash bucket. This is the only place I've seen this.
In Greece, dishwasher again. The sink is also split like in Sweden.
In Japan, washing dishes in a single large sink under running water.
Edits:
* The UK also has those braindead hot and cold in separate faucets on each side of the sink. So when you wash your hands, either you're wasting a ton of water, you're using cold water, or you're scalding your hands. Will never understand that. And this is in a NEW apartment!
* Also, many brits I've known don't rinse their dishes... they let them dry with the soap on them. Also inexplicable.
Depends on the sink. If you've got a sufficiently large countertop, it's not uncommon to have a steel sink comprised of two basins, one for washing/soaking the other for rinsing.
Older construction with porcelain or enameled sinks often have a single basin. Both this and the finish make a removable plastic washtub useful both to protect dishes and allow for rinsing whilst washing. Some "efficiency" apartments may also have smaller sinks for various reasons and only offer a single basin, again, a washtub can be helpful.
The few I've seen have been people with only one sink (so, a single large depression), which makes it easier to wash and rinse. But if it's not large enough, none of them fit, so it's mostly an exercise in futility.
And yes, dishwashers are absolutely everywhere. Some don't have them of course, but they're standard fare in a huge number of places, even relatively cheap apartments.
Also, my kitchen sink is actually slightly slanted towards the sink hole.