Hi! New S40 rider here. Also largely mechanical newb.

Bike is a 2012 model, 7400 miles (bought at 7100 or so 2 months back). No modifications. In the Piedmont of North Carolina (400ft), so currently a few degrees above freezing most mornings.

Since getting the bike, I've had an on-and-off issue of it stalling when I dethrottle to idle. Always starts just fine with the choke in the morning, warms up and idles fine.

Some days, that's it - runs beautifully. Some rides, too - even after a bad day - runs beautifully.
Most days and most rides, though, I'll get anywhere from one to three stops down the road and when I dethrottle to stop, the engine idles down too far, stumbles and stalls. Sometimes, but not always, there will be a smell of unburned gas while riding, and it will always stall at the next idle when this is the case (too rich mixture?)

When this happens, gas will leak out the carb overflow for a minute or two afterwards. The engine will start again without choke if I roll the throttle on about 1/4 OR run the starter for 6-7 seconds, then will usually idle high for a minute then be ok. But then, two stops down the road, it's stalled again and dribbling gas. At present, will start without choke or throttle, but will begin leaking gas from the carb overflow as long as the engine is idling.

Given the carb overflow gas leak, presumably the carb is getting flooded. However, turning the petcock to PRIME either before, during or after a stall does not cause any change in the behavior - a stuck float valve would be the obvious culprit, except I'd expect a stuck valve to leak with the petcock on prime...

I have also tried adjusting the idle speed, which seems to make no difference. Tapping the carb with a rubber hammer doesn't help.

The dealership it was bought from have taken it back in twice. First time they stripped, cleaned and reassembled the carb, and the bike ran fine for them for 3-4 miles, but stalled within 200 feet of my house when they brought it back. Second time, they put 15 miles on it without being able to replicate the problem, so I think they just gave it back to me without doing anything (annoying, of course, that one of it's "running like a dream" journeys would be when the tech was riding it...). It's going in a third time for 7500-mile service in a few days, and I'd love to give them some clear pointers on specific things to check, or things I can try in the meantime.  

Any ideas?

I do still have the original petcock (I plan to pop a raptor on there, but given that I've now owned my first bike for, what, two months? I'm hesitant...) - would that cause gas to flow out of the carb overflow line?

Alex , your saying carb overflow line ,there are two lines that are tucked up into the backbone of the frame .they should both leak fuel if the bike is upright .are these the lines your talking about?......I think you have a small hole in the rubber diaphragm in the petcock,yes you've only 7000 miles on the bike ,but it's 5 years old which means it sat around a long time with old gas in the petcock . buy a raptor ,change it yourself! buy an adjustable wrench! skip the dealer! they no very little about the care and feeding of the S-40!

Can you take a photo of the tube you are talking about....does it terminate just behind the engine/front of the swing arm?  It could be that the tube is the drain for the air box.....and your fuel is flowing into the air box when the petcock vacuum line runs fuel into the carb....or the float bowl overflows.

If that is the air box drain....it is not supposed to be a vent or overflow tube - it is the place where un-burned oil from the crankcase vent collects.....and you are supposed to remove the cap and drain the "spooge" when you do an oil change.

Dave and Kenny ,are right on! but if you have gas running out this tube it could also be running into your cylinder and down into your oil,I would hold off riding the bike until you change the petcock (drain tht gas from the tank by using "pri" then unbolt the petcock,replace with raptor,it has the same base) then plug the opening on the carb where the vacuum line was,(you can buy rubber caps at any parts store) then check your oil, open the fill cap and if you smell fuel in your oil you need to change it and filter ,you might want to change your air filter as well. If it was suggested to leave the cap off that drain hose by someone at the dealership ,that is a perfect example of them not knowing $hit about your S-40 ! If after you change this you still have the problem ,you have a problem with the carb float or float valve it can't be anything else.

You should also replace the plug on the air box drain tube,it should not be open all the time,if it is open when the motor is running the carb will pull in air through it and it enters the air box between the filter and carb(dirty) and ruin your motor over time.

A gasoline soaked air filter really messes with how it runs.
If you're getting that much gas coming from the carb going into the airbox that it's showing up on the floor,, you REALLY need to fix that and you probably have gas in your oil.

That "ring" on the drain hose is a clamp used to hold a plastic plug ,but a small bolt would work as well.