We are having this same problem. We had a new kitchen faucet installed about 6 months ago - single lever - and everything was great until a few weeks ago. Then it started. When we turn on the cold water, the first blast is soapy. After we let it run a few seconds it seems to clear out. What could be causing this? (We are on city water, and we have always had soapy water come up from the drain when we are using the dishwasher or the washing machine.)

we have always had soapy water come up from the drain when we are using the dishwasher or the washing machine.) Remove the aerator and tell me if soapy water still comes out the spout.
You have a partial blockage in the horizontal line in the wall. The solution would be to remove the traps "J" bend and send a snake up into the wall,(see image). There's a bend ablout 8 inches in but once around that you only have to put out 6 more feet of snake. Flush with a tub of hot water. Good luck and thank you for ratting my answer. Tom