When I run water in my bathroom sink and then put the plug in whilst there is still some water left in the basin, there is a high pitch moaning or screeching noise made by my adding the plug. What causes this noise?
We tracked a similar sound to a tiny leak in the flush/refill mechanism in the toilet. Any brief alteration of water pressure anywhere in the bathroom could trigger the sound. If you still get that sound when using a sink plug that you know for sure does not leak, try flushing the toilet to see if it stops the sound (even though the sound seems to be coming from the sink).