Is your basement just for storage or living in? If the latter, damp proofing, thermal insulation and ventilation are preliminaries that ought to be dealt with before you can make your choice of flooring.
If you’re converting a cellar into habitable rooms and lowering the floor level to improve the ceiling height, planning permission will require damp-proofing and insulation anyway, but the rules are currently under review.
Timber Floor Boards
There is no type of wooden flooring that’s happy with dampness if you haven’t got the sub floor sorted, but traditional floorboards above treated joists will hold up better than most. They are cumbersome to fit properly in a tight basement space and changes in humidity and temperature can cause shrinkage issues later.
Engineered Wood Flooring
These boards are lighter, cheaper, and easier to fit. Sound insulation has sometimes been an objection to this material but this isn’t an issue in a basement. Check out a range of designs from companies such as http://www.ukflooringdirect.co.uk/engineered-wood-flooring. A light glossy colour will brighten a basement with little or no natural light.
Only a few types are approved for use in damp areas like bathrooms and kitchens because they’re not too stable in reaction to either temperature or humidity. They’re designed to float on a sub-floor (you could use ply, for example) which must be very flat. On the other hand, because they float they’re easy to replace if needed.
For floors that aren’t damp proofed your options are usually reduced to ceramic tiles, bricks or concrete. Concrete can be polished to be attractive but will always be hard, cold and potentially prone to efflorescence unless sealed. LVT (luxury vinyl tiles) are resistant to damp and warping, and would ameliorate that hardness but see the warning below. Engineering bricks impregnated with bitumen will resist damp and might allow you to lay carpet or other options above them.
As ventilation can be an issue in basements, one material to avoid is vinyl – either in rolls or tiles. This material is inadvisable for its tendency to slowly release formaldehyde and other toxins, and you don’t want them building up dangerously in an area where ventilation is limited. To a lesser extent the same warning applies to many types of carpet.