The drip comes from the heating/ac vent in the basement ceiling about four feet from where the kitchen sink is above. The area under the sink in the kitchen is dry. The drip only occurs when the kitchen sink has been running but doesn't start until some minutes have passed and lasts after the water has been turned off. About a 1/4 of water accumulates in the bucket I have catching it, so some water is escaping in the waste pipe, but some is leaking from the pipe. Any thoughts on how to find the actual leak?

Thanks!
Angela

I meant a "leak". :)

The only real way is to cut an access hole or two in the ceiling and look for the leak.

Is it more likely to be at a joint? Or can the pipes rust through at any point along the way?

If the drain line is cast iron it can leak anywhere. If plastic -PVC -it is usually at a joint but could also be a crack. Lots of possibility so that is why your need to open the ceiling.

Thanks very much. It is definitely a cast iron pipe. The house is 100 years old.

Rather than trying to patch it, replace with PVC.