Doesn’t the overflow drain on a bathroom sink, for example, have to connect just upstream of the P-trap (to prevent sewer gas from entering your home)? And if so, don’t most clogs form at the P-trap? And, if so, how does the overflow drain gain you anything??? Please explain how the overflow drain can work, although it seems so innocent and obvious on the surface. 
The overflow drain is built into the sink and drains into the drain pipe just below the drain fitting in the basin. If your trap is clogged it does no good, it’s to prevent overflow in the basin when the drain is closed. If your trap is clogged then clean it out and this won’t be a problem.
Mine dumps directly on the ground.
Well that’s better than nothing because if your sink clogged up you’d end up with water all over the ground. It’s almost as useful as putting a drain pipe without the overflow openings into the sink.
If you mean , down through the floor or wall to earth , then I guess thats alright…
But if you mean to the floor, then thats really not helpful as an overflow… The hose may have been removed from there to act as the drain for a machine ??? Or perhaps the hose was never installed (no overflow tub, the cap to the hole is missing.)
If the overflow did connect to below the P or S bend, then bad smells could flow up through the overflow… That is because the water flowing down pushes air up… the bends create enough pressure that the air flows up the vent instead of through the drains…
The purpose of the overflow drain is to prevent a spill when you’ve got the the stopper in the primary drain. It doesn’t do anything for you if there’s a clog in the P-trap. If that’s the case, it’s time to go get the snake.
Make sure your mouth is closed when you pull the snake out of the drain … a professional tip here.
You gains you freedom from running untold gallons of water onto the floor if you happen to leave the faucet running while the normal drain hole is closed. It’s not meant to do anything relative to a clogged P-trap – apples and oranges there.
Again: It is to relieve the possibility of overflow when someone starts to draw a basin full of water, then forgets about it.
Or kids screwing around.
Your bathtub also has an overflow - usually on the underside of trap lever.
Which is why kids screwing around will stuff TP into the overflow before plugging the drain, turning on the water, and leaving the school restroom.
I’ve heard.
Dem pipes dem pipes dem wet pipes.
In my case:
Bathtub: The overflow goes straight down to the trap. The drain joins it from the side. Replacing this was a … job. The flip stopper inside it didn’t work anymore. It was a tight space but at least I could access it from underneath via the garage.
Bathroom sinks: There are slots in the drainpipe near the top. Between the top and bottom surfaces of the basin. The overflows drain thru a “bulge” in the basin on the underside to the drain opening at the bottom. It’s important when installing a new drain that you don’t block the slots by going crazy with the putty.
Kitchen sink: No overflow. But it is a double basin with the partition top lower than the rim.
Gross
But true
If you look at the underside of the sink, the overflow is a ceramic(usually) channel stuck to the outside of the bow and leading to the drain. On my sinks, there’s a visible ridge about an inch high two inches wide leading from the overflow drain to where the regular drain attaches. The regular drain has an opening between where it seals on the bottom of the sink and where the flange on the top of the sink drain attaches.
Makes sense. Thanks for the explanation!
There are some kitchen sinks with overflows. My European Blanco has one. It is a fairly complex molded sink with a drainboard along one side also.
Dennis