Our basement is usually dry, but every so often we have torrential rainstorms in the spring and we are downhill from our neighbors. We get some water overnight in a small stream that we can mop up. We had carpet with pad, but had to have it taken out this year. Would sheet vinal dry if a puddle of water came up from the concrete under it? I can mop up the small surface stream that seems to happen every few years, but I'm concerned about the effect of any water that might come up under the vinyl floor. Would vinyl tiles be better as any puddle that came up through the concrete floor would be able to escape through the cracks between tiles? Would the good vinyl tiles withstand overnight water and dry without damage?
Before you jump too far, you should figure out the exact source of the water. When you're talking about puddles and streams, I'm thinking leaks and cracks, and not moisture coming through the concrete.
To check for moisture in the concrete, tape a large square piece of clear vinyl down using duct tape (at least a foot square) at a time when the floor is very dry. Leave it for a while, and if there is any water or moisture in it, you'll know it's coming through the concrete.
The problem is that if you have any amount of moisture coming through that way, anything you stick to the floor (like vinyl tiles) will be disrupted by it: it will get in the adhesive from the tiles, and they'll eventually all come unstuck.
Another source of water is cracks and leaks around windows, the joint between the floor/walls, etc. On a rainy day, you may be able to see water coming in, and you may just be able to spot these (especially cracks) visually. There are different ways to fix these problems, depending on exactly what it is.
Yet another common source is by an overflowing sump pit, which is just what it sounds like it. If you have a pit, and your pump is not working, or can't keep up, obviously that water has to go somewhere. It's also a very good idea to have a secondary battery-backed pump if you have a basement prone to flooding. Since you didn't mention this in the question, it's probably not the problem.
Assuming you have fixed any all-out leaks, and are just dealing with water wicking through the concrete, a product like Subflor or DeltaFL (available at any big box stores) would probably be the best choice, if you're really intent on having some kind of flooring (eg, using the basement as a finished room). It's basically a plastic membrane with dimples in it, which leaves about 3/8" of an air gap for drying, and also provides some insulation for the floor above. Subflor comes in 2x2' tiles with OSB attached, and DeltaFL comes in rolls, and you put plywood or OSB sheets on top, or you can install hardwood or laminate directly on top (deltaFL also acts as a vapour barrier).