The default really is white linen. I've found the Brooks Brothers three pack to be quite useful. I believe Four-in-Hand may offer white linen squares in singles. He does offer some with colored edges, white linen with blue edging is also very versatile. I don't believe that linen squares are generally regarded to have any seasonal limitation.
If you're spending money on a pocket square make sure the edging is handrolled. I posted this in an earlier thread to show the difference, the two on the left are machine stitched, the ones on the right are handrolled:
Notice how the handrolled ones are plumper and rounded on the edge. David shows well how this is done on the Sam Hober website that he linked. I fold machine stitched squares so the edge doesn't show, something like this works well:
I've moved away from solid silk squares, largely by the demonstration by cuffthis in his old daily pictures of how patterned silks add so much to the ensemble. An off-white silk square can be useful, though.
Take a look at the Sam Hober site for some drool inducing wool squares, which keep calling out to me.
For a good source of cheap patterned squares go to TJMaxx/Marshall's, and you'll find in the men's section 3-pack boxes of patterned cotton handkerchiefs from brands like Nautica. Find a pack with patterns and colors you like and fold them so the machine stitched edges don't show. These can work really well with tweeds and also blazers.
Finally, never exactly match your pocket square to your tie, and take a look at this guide at M.S. McClellan
on pocket squares.