Get thee to the John Bridge forum! Lots of good information from professionals there. They are a great group, more than willing to help.

Just briefly, though: Kitchen counterop - If you want to tile to plywood, first make sure you have adequate plywood thickness (at least 1 layer 3/4 plus one layer 1/2") then Ditra membrane, then tile with thinset.

No tile over vinyl flooring, unless you enjoy tiling so much that you want to remove it and do it over when it fails.

Bare cement - entirely depends on the condition of the cement.

Okay-- one at a time:

Is applying Versabond directly to plywood (kitchen counter)

This can be done, and in fact it's the way I build my countertops-- two layers of 3/4" ply. However, I don't recommend going directly over plywood for DIYers, and in fact, many pros, as well. To start with, Versabond, although it says on the bag that it'll stick to plywood, it won't last very long. You need something with a pliable and tenacious bond, such as Mapei's Kerabond thinset mixed with their Keralastic additive. Additionally, there are many other pitfalls in setting directly over plywood and it's not something I'd really recommend unless you're very familiar with what you're doing. You'd be much better off to instead, use a layer of 3/4" ply, a layer of 1/2" ply, and then either Ditra or 1/4" cement board. You definitely want two layers of plywood underneath, though.

bare cement (basement floor)

As alice siad, it depends on the condition. If you have alot of cracks in the floor, you may want to put a membrane down first, such as Ditra. If there are only a couple, you might get away with just using a crack supression membrane over the cracks such as Laticrete's 9235 (Blue 92). But membraning the entire floor would be good insurance. One very important note, though-- if you DO have cracks in the slab, check them very carefully. if on ANY ONE of those cracks, one side is higher than the other, you might want to go with a different flooring. There's not a membrane made that'll protect tile from vertical movement between the two sides of a cracked slab.

or existing (tight) vinyl tile (kitchen floor)

This is copied and pasted from the flooring page of my website:

This is a subject that's very hotly debated within the industry. It seems every manufacturer has a thinset that they say can be used over resilient flooring (sheet vinyl) or vinyl tile. The problem with that is twofold. With all the different types of vinyl flooring out now, it's tough to know which ones can, and which ones can't be tiled over. Cushioned vinyl's are out. They can compress, causing voids under the tile, ultimately causing failure. Also, not all sheet vinyl is glued solid to the floor. A lot of times, vinyl installers will only glue the edges and spot glue in the center, and if you thinset the tile to it, it won't stay very long. In my opinion, vinyl tiles are out, as well, due to the fact that the thinset bond is only as good as the bond of the surface under it, and I've seen vinyl tiles ( especially peel and stick) let go way too easily. The same is also true for sheet vinyl. The second problem with tiling over vinyl is the underlayments used for vinyl installation. Normally, in woodframe construction, there are one of several underlayments used-- luan, 1/4" particle board, 1/4" plywood, and sometimes even MDF (medium density fiberboard). ANY of those in a tile subfloor is a guaranteed failure. 1/4" plywood has a nasty habit of delaminating (the layers come apart). Luan has the same problem, plus it can compress to 1/2 its original thickness from normal residential foot traffic (I've seen this happen). Particle board (as well as MDF), just from humidity, can expand enough to pop tile loose. Any of those can cause failure, and unless you pull the vinyl, you don't know for sure whether the vinyl installer just went over the existing subfloor, or added one of the underlayments mentioned. All in all, it's an extremely risky installation, and my question is why, when spending the money it costs for a tile installation, would you want to gamble on it?

Here is a link that might be useful: My website's floors page

Flexbond is more modified than Vesabond too.

I still wouldn't use anything over plywood less than a good unmodified thinset mixed full strength with a liquid latex additive. I've seen too many modified thinsets let go. A good friend of mine owns a thinset company that back a few years ago came out with a modified thinset specifically meant to compete with the likes of K/K. He asked me one time why I wouldn't use it, and I told him the same thing. Until it's been out for a while and I've seen it hold on, I won't even consider it.

Thanks to you all for your suggestions. I now have professional advice, in writing, that I can show to my husband and hopefully he will listen!

I agree, I think that kerabond w/ keralastic is about the stickiest stuff out there.

I threw out the Flexbond name because the O.P. mentioned Versabond.

And I just want to say "good words" regarding the tile over the slab. Even if the slab is in top-notch shape, a basic isolation membrane help when you get differential expansion of a below-grade slab and tile.

Oh, lordy, my timer is going off which means my contact cement had flashed over. Time to get back to work. I'm installing several hundred feet of cork on walls, and the contact cement fumes are...pungent, to say the least. Four gallons of contact cement on this job. Wahoo!

And this is "old school" contact cement. None of that eco-friendly crap! 8^)

Mongo