https://www.warmlyyours.com/en-US/floor-heating

You can use a electric radiant floor heat and have hardwood or tiles installed over it. Check out Thermosoft.com

Vinyl planks? The fancy ones? Then buy an inexpensive rug to put on top.

I live in the Midwest where it gets pretty cold in the winter. We just finished our basement and put LVP there (floating). We didn't want the chemicals with the adhesive so that's why we opted to float it. When it was unfinished it was just concrete and freezing. Now you can go down there barefoot. I'm amazed at how much warmer the basement is just by finishing it.

We used Coretec that has a cork layer like SJ McCarthy suggested, but we did not add an additional cork layer, just the floating floor on top of the concrete and it's still pretty warm.

If the height is available, consider 1"-2" of extruded polystyrene (XPS) followed by a layer or two of ply or OSB. The XPS is impervious to water--we use it for floating docks--and provides R5 per inch.

Take a look at epoxy floors. Not the garage floor kits from the home store, but professionally installed epoxy. I put it in our basement family room and am very happy with it. It’s much warmer underfoot than the plain concrete and no worries about water on top of it. If your water issues are coming from below then it won’t be a good choice. I believe the brand used was Reflector. We recently had a plumbing leak and the floor part of cleanup was a breeze.

Since my original post, I've had another water event in that basement. We've abandoned the idea of radiant hot-water heat under the floor because of installation difficulties and expense, although running baseboard hot water along an additional wall may be a useful option that won't break the installation bank nor cost any extra to run. It's also become apparent that the floor surface and perimeter need to be sealed to prevent any water seeping and pooling under the floor. Epoxy isn't insulating enough and area rugs aren't large enough to be useful.

We're looking at some type of system that incorporates insulation over the concrete, electric radiant heat, and tile. My contractor is checking out Schluter-Ditra and similar products. I'm trying to find out how much this is going to impact my electric usage, and if anyone has any links to info about those costs, I'd be grateful.

Which area of New England are you located in? We serve eastern Mass.

What is causing the "water events"? Is it due to EXTERNAL water (rain, puddles, poor gutters, badly sloped lot) or is it internal plumbing issues? Until you fix the BIG issue, the type of floor you put in won't matter. It will always need replacing until the real concern is dealt with.

And to be clear, the electric systems can be expensive to run if you do not have the insulation integrated UNDERNEATH the slab. A retro-fit can be very expensive to run. And the electric stuff HATES to be submerged in standing water (like a small flood from a 'water event'). The electric systems can be shorted out if they are exposed to standing water.

G&S--I'm in northern NH.

Winnepesauki area? All the snow is melting and causing water issues throughout the area, including Mass. this time of year. If, you have the head clearance, I suggest installing a subfloor on sleepers, if, you are only getting inches of water. That will at least allow water to stay below the actual flooring or interior french drains.