Plywood is a stable base that works pretty good if sealed. Problem is the edging. It looks like plywood and even the best has voids. So for an attractive base, you have to use molding, edge tape or a picture frame to give it a nice finish. The thicker the plywood (more plys) the heavier and more stable it will be. While it is easy enough to find 3/4" molding to use along the edge, anything thinner becomes difficult to readily find. Since I have a pretty complete wood shop, I can make my own, with what ever profile I like and what ever thickness. (note: 3/4" plywood isn't exactly 3/4", there is always a little lip of molding left over, but this can work for you. As was stated, if you don't seal it and use a wet mix of celuclay it will warp or worse, delaminate. Marine grade is nice for this, but at $50-$65 for a 4x8 sheet, expense. Unless you use a plywood blade or carefully score the cut line you will get some tearout. Another detraction along the edge. Because you can find it in 4x8 sheets you can make any size you like. If you don't have the means to cut it, most home centers will cut it for free. Not real accurate, but to within 1/4". The "handy panels" are priced accordingly (no cheaper, or more expensive, just handier because they are ususally cut into 4x4, 2x4, 2x2 and are easier than getting a whole sheet.
Solid wood is nice and makes for some attractive bases. Board lumber is a happy thing though and can suffer from humidity conditions and go so far as to warp, split, check and all the other head agaist the wall banging associated with it. Grain movement can pop ground material loose and warp if it gets wet and wasn't sealed. In this case, both sides and the ends must be sealed, well. Again, well. If you only seal oneside (like you could do with plywood) the wood grain movement on the unsealed side will not be equal to the sealed. Resulting in warping. Also there are 2 sides of endgrain. More of an aesthetic issue than anything else but can make for some odd finishing problems. Again, I emphasise if you are going to use this, seal it all around. In most cases, unless you are going to pick up an edgeglued panel or premade table top the widest you'll find is 11.5" wide (a 1"x12")
MDF or medium density fibre board is just that. It is a fine wood and paper pulp, that is compressed and made into lumber. Not to be confused with particle board or melanine it is a very easy material to work with and looks great if treated. Its thickness is 3/4" and comes in thickenesses from 1/4" to 1" depending on your supplier. It is cheap and can be cut to any size. Or is avaialble in precut handi-panels. THis stuff machines well and is heavy, making a very stable base. It desn't suffer from wood movement or endgrain. It is finish smooth on two sides and with a little sanding can be made perfectly smooth on all sides and edges. Its advantage is that it is cheap (a 4x8 sheet is anywhere from $10-15 for 3/4"). Once primed it can be painted to look like anything. Rock, steel, plastic, wood, what ever. Faux painting these is half the fun. Because of their weight they are also bottom heavy and a smaller piece can be used to provide a stable base for a top heavy presentation, lets say like an aircraft in flight or a figure in an extreme pose. They machine easy so a variety of edge treatments can be done. It must be primed for the best finish though and at the least must be sealed. As if water hits an unsealed surface it will blister as a result of the wood and paper pulp or fibres swelling up. They can be sanded smooth if this happens for an invisible fix though. The dust is pretty noxious though because of the binders and its fineness so I would reccomend a dust mask and goggles if you work with it. I have even noticed a sensitivity to skin, but that may be me.
Other base materials that I've seen used with great success are tiles, wooden plaques from craft stores, trophy bases, plates turned upside down, field stone, slate, a log (you had to see it, it was the perfect base for the figures) caps from various paint cans, jars, or containers, finials, and I think its pretty much safe to say just about anything that will not move and will allow you to fix a figure or item to it.
Mike