Hello Dave, welcome to the Cameralabs forums!
To answer your question you have to ask yourself at what distance the prints will normally be viewed and what quality you'll be satisfied with. Obviously the further away you are when viewing, the more forgiving you'll be.
So-called 'photo' quality prints, such as those you get from labs which are designed to be looked at from very close range, are typically printed at 300 dpi. This means you'll need 300 colour pixels per inch.
To work out how big a print you can make from a camera at 300 dpi, just look-up it's maximum horizontal and vertical pixels from our Cameras menu, and divide each by 300.
So for example, the Olympus E500 has 3264 x 2448 pixels, which when divided by 300 would make a print measuring about 11x8in.
That may not sound that big, but remember it's printed at 300 dpi which means it's designed to be studied from only a few inches away. Stand back a couple of feet and a print would look great at 200 dpi, where the E500 for example would allow you to produce something measuring about 16x12in.
Obviously there's also higher resolution cameras than the E500 now, including many 10 Megapixel models which will give you an extra inch or two.
Ultimately it all depends on your expectations - and the printing process used, as some are quite forgiving with lower resolution images. It's just personal, but I find inkjet prints can look great even at quite close range when printed at 200 dpi. I've also made several A3 sized prints from 10 Megapixel images which I reckon look pretty good.
I'd advise getting hold of a 10 Megapixel image from a friend or perhaps a shop which will let you fire one off in the store, then try printing it at different sizes to see how well it holds up. You may find it's absolutely fine printed at 24in, or maybe intolerable. Only you (and your clients / or application) can decide.
If it makes you feel better, a good 35mm film scan would only contain about 10-13 Megapixels worth of usable detail - at best - so, the current crop of DSLRs are very very close to what that can achieve, and the better models already exceed it. So if you've seen 35mm blown up to 24in and think it looks ok, then a modern DSLR should be able to match it.
(and if you've seen those massive 'Earth from the Air' exhibition prints by Yann Arthus-Bertrand, remember they're mostly made from 35mm film and look pretty damn good!)
Note I have specified a DSLR though. While many compacts now offer 10 Megapixels, the images from a DSLR are much cleaner and better suited to bigger prints. There's some great 10 Megapixel DSLRs available right now, including a new Olympus model if you're loyal to this particular brand, so I'd say your first step is to make a large print from a 10 Megapixel file to see if it measures up. I'd also recommend using a quality lens, as large prints will show-up any problems on cheaper models.
If it look ok, then great! If not, you'll need to spend considerably more on a higher resolution DSLR such as the 12.8 Mpixel Canon 5D, or maybe even the 16.7 Mpixel 1Ds Mark II (although be aware this is likely to be updated in the next few months).
Or if these aren't good enough and you want 300 dpi quality on a very large print, then a significant investment in medium format equipment may have to be the way forward. But to be honest, I think most people would be very impressed with what you can do with a typcial 10 Mpixel DSLR file, especially if the lens quality was also good.
Let us know how you get on,