What should I look for in strawberries at the market?

I mostly agree with Rumtscho, except that I think "red through and through" is a bit too strong: a bit of white at the top doesn't have to mean anything, as long as it is not too much, and as long as there is not white in the bottom half—so these look good:

Another thing to watch out for is how far apart the seeds are. I think one can never be sure of anything with strawberries, but, if the seeds are farther apart, that usually means they are sweeter. I also like the ones where the seeds are a bit deeper into the skin better. The ones above look good in both respects. These, however, do not:

Smell. Really, this is the most reliable criterion for practically any fruit.

Flavor contains both taste and smell. For a strawberry, you want a fragrant smell together with enough sweetness. For both, the berry has to be ripe enough. If it was picked underripe, it won't smell good enough yet, and it will also be hard and sour. If it was picked long ago and is not fresh any more, it will have lost the more volatile components of its fragrance. So ripe, fresh strawberries smell great. I have had occasions when I entered a supermarket to quickly buy one thing, aimed at the correct aisle, but when the smell of good strawberries reached me near the produce, I turned and added a pack of them. I have never been disappointed with such strawberries. Also, if you smell the slightest hint of mold, fermentation, or foulness, you know they may not keep even one night.

Another sign is that a ripe strawberry will be red through and through. A strawberry picked underripe will be white or even slightly greenish at the top. It doesn't taste good then. But this is a negative sign, because not all red strawberries taste good.

Don't ever go by shape. The tastiest sort of strawberries my grandparents grew produced ugly, lumpy strawberries of a light, slightly orange color. They also had a few rows of a sort which produced perfectly conical, deep red strawberries, they looked like an advertisement - but they were hard and dry, and didn't have much aroma. Probably, there are some strawberries which both look and taste good - just don't think that looks or color predict a good strawberry, because they are independent.

And of course, any strawberries which have visible mold or fouled spots are not good any more. You can usually also recognize overripened strawberries in their appearance, but I don't know how to describe it well. They just look old.

One thing I learnt from villagers in Central/Eastern Europe where wild strawberries were often seen growing along the sides of streets was that smaller strawberries have more flavor. It was though each strawberry was born with 'x' units of flavor, so the bigger it got, the less flavor-dense it became.

I've never done any scientific testing of this, but I must say since those days I've been only choosing smaller strawberries and my success rate feels like it has gone up.

This is anecdotal, but I think accurate. I am not a farmer, but buy strawberries at a farmer's market where there is quite a bit of friendly competition for things like "best tasting strawberries." From what I've heard, farmers here make a choice whether or not to irrigate their berry plants. Those who do tend to end up with bumper crops of not very exciting strawberries. Those who don't have lower yields, but more intensely-flavored strawberries. You can always ask the farmer, but it doesn't take much taste-testing to figure out which ones go for the concentrated flavor over a quick buck. I'm not sure about the science behind this, but I've heard it holds true for tomatoes as well.