Faucet needs repaired.
old pipes in the sink or tub upstairs
Is the water not hot? Or no water coming out the hot tap? not comming out at all
A blocked pipe. The water can not go down the pipe, so it comes out in your sink.
There is a clog between the sink and the line to the street or septic tank. The sink is the easier path for the upstairs shower.
This could happen due to improper plumbing, or due to a clog. If the waste water is coming up through a sink and not a toilet, it's more likely that an upstairs toilet and a downstairs (or lower) sink share a waste stack that is clogged below a shared junction between the two.
Cold water line could be rusted shut or clogged. Valve supplying the fixtures may be turned off.
From dirt water
You cannot do this without a way of stopping the water. The sink sprayer is hooked up through your kitchen faucet for a reason. When the faucet is shut off, no water can come through the sprayer. If you hook up the sprayer to a water filter, the handle on the sprayer cannot handle the pressure of the water system and will continually run when the handle packs it in, more than likely when you are not home!
There is a main drain that runs from the basement through the roof where it is the vent for the system. If you look at the layout of the house, the upstairs bath is most likely over the downstairs one or the kitchen. If you look on the roof, you will see a 4 inch pipe coming through the roof that should be over the upstairs bath. The sink and shower drain into that and then to the basement and out of the house. Most likely the main stack is in the wall behind the toilet.
Unscrew the aerator from the end of the faucet where the water comes out and clean the screen. Sediment build up slows the flow.
Dishwasher water may come out of the sink if there is a clog in the drain pipe that connects the dishwasher to the sink. This can cause water to back up and overflow into the sink. It is important to address this issue promptly to prevent further damage and ensure proper drainage.
You need to turn it off at the base of the sink. Or depending on the problem you may need to turn it off at the water main valve.