Found my first Suzuki last week, an 06 S40. Guess the seller dropped it on her knee in the drive and gave up riding after that, so it sat.

Realizing it'll never be perfect, I'd like to hear suggestions for popping this back out. Attempted to add a photo, but due to this being my first post, guess I can't. Will see if I can reply to this with a photo. thanks

As long as there are no actual creases, I've heard dry ice works really well.  Drain the gas, freeze the tank around the dent, up to a couple of inches around the dent.  Then watch as it warms up and the dent comes out!  

dry ice?
suction?
time machine?

I have heard a plunger for dents in car. Not sure if it will work on the bike...

Jerry the dent doctors in most cases have to get behind the dent to work the steel. Todays cars an light trk.s have rather thin sheetmetal and are really not that hard to repair our shop uses them offten and they do very good work price not to bad eather. As for the tank dent he might want to stop at a well equiped body shop some have a spl. tool that hooks to a slidehammer then to a vac. supply the amount of suction and the pull from the slide hammer pull some pritty nasty dents out. It's worth the trip to check it out.You may luck outand one of the body guys may be a biker mite do you a favor.
Ride on Ride safe
viper

Iheard using air would make it leak, maybe even split a seam. Pressing it out with a round piece will screw it up royally. A relatively flat piece with softened edges woud help. I would consider taking it to a paintless dent repair place & just talk to someone. Maybe if you pull the tank & air it out a couple of days, so a spark wouldnt be a disaster, ( I dont know if that will work, is 2 days enough?) Maybe someone could go up thru the petcock hole & spank that dent out. Your assessment of how it got there seems reasonable. Mine is bent there, too. I was walking away,parked on a downhill( Dummy!) & it was rolling off the stand, jumped back at it & stck my right leg under it. Lucky I didnt wreck my knee again. Its been operated on once. REflexes dont run thru the "Is this a smart thing to do" part of the brain, tho.  Apparently, from looking at a lot of my posts, I do a lot of living on reflex.

If you can get a flat bar that will get between the feet on that thing & wrap the bar with a soft rag & very, ohh very lightly spank on the bar, worink the "eyebrow" down as the bottom of the dent comes up. 100 light taps is better than 5 smacks. Its okay if you cant see the metal move, be patient, A dent like that has a high ridge around it, sometimes just on one side, but thers one there somewhere. Just make sure all adges are rounded off. You can even use wood. Still have to wrap it, tho. I wish you the best.