Where the water appears, and what else is affected, narrows the blockage down to one of about four places before anyone opens anything.
If one side backs up and the other drains, the blockage is in the branch serving that bowl, upstream of where the two join. That is the shortest and cheapest kind of blockage to reach, and it is frequently in the trap directly under that bowl.
The blockage sits after the two sides join but before the kitchen line meets the main. Grease is the usual culprit in a kitchen line. It goes down warm and liquid, cools within a few feet, and builds on the pipe wall over months until the bore is a fraction of what it should be.
The dishwasher shares the kitchen drain. If it empties into a backed-up line, the water surfaces wherever it can. Check the air gap on the worktop if the house has one, and the high loop in the dishwasher drain hose — a hose that has slipped down below the trap level causes exactly this and is a free fix.
If the kitchen backs up and a bathroom fixture is also slow, or a floor drain or toilet reacts when the kitchen drains, stop treating it as a kitchen problem. The restriction is in the main line serving all of them, and clearing the kitchen trap will not touch it. That is a camera job.
Cooking grease and oil, obviously, but also coffee grounds, which accumulate rather than break up, and starchy waste like pasta and rice, which swell. A disposal does not change any of this — it reduces the size of what goes down, not the tendency of grease to coat the pipe.
Sometimes, on a soft nearby clog. Block the other bowl's drain first or the pressure simply goes there. It will not shift a grease-coated line.
Occasionally — it can jam or its own trap can block. If the disposal hums without turning, that is the unit. If it runs normally and water still backs up, the problem is downstream of it.
Typical of a grease-narrowed line. There is enough bore for ordinary use until one heavier load overwhelms it. Clearing the coating rather than the plug is what fixes it for good.
Send the details and we'll route it to a licensed local contractor.