Noticed this morning that we could hear water constantly running in sewer pipe. Shut off all appliances and toilets and main water to house. Still running. We are in midst of extremely heavy rains. Could enough rain water be flowing down the multiple vent stacks on the house to make this flow? No evidence of any leaks into house.

It is possible, water is falling into plumbing vents on the roof and slowly running down to the drain. Also, your toilet may have small leak from the tank. Moreover, it may be condensation run off from your A/C or Reverse Osmosis discharge. Lastly, it can be dripping faucet. Check on all listed

Back to you

Milo

If the house water is off and the toilets don't leak, there is little other possibility than rain water. You would mostly hear this with PVC drains. A 2" vent would allow a lot of water down it, look at what collects in a rain gauge. You have a sump pump?