I'm voting for leaf miner, apparently there is one that is particularly fond of spinach, beets, and chard and the damage looks very similar. http://anevidencebasedgardener.wordpress.com/2011/06/16/leafminers-attack/

Leaf mining injuries are also rarely fatal to the plant as a whole. While you probably wont be eating any of those leaves, I'd lop them off, trash them, and let the chard continue growing.

I found only way to deal with leaf miner is covering the plantings with insect barrier or tulle. I lost whole beets crop due to that miner 2 years in the row. Now I cover them as soon as I plant them and they grow OK.

Severe leaf miner infestation.

Cut off and trash/burn all affected leaves.

Spray remainder of plants with Spinosad (aka Monterey garden insect spray) to kill any remaining larvae that hatch, and cover immediately with insect-barrier grade row cover. Take the cover off only to weed and harvest.

How much chard do you have? Slimy's solution seems like overkill (no pun intended) to me if you only have those 3 plants. Even if they were all healthy and you picked and cooked every leaf in the picture you'd have about 2 mouthfuls of chard. I'd ditch these and sow a LOT more if you want a few portions to eat.

BTW your lettuce back right needs to be eaten today. It's bolting, as I'm sure you know. The lettuce bottom right is also on its way. Maybe mix the lettuce with the remaining good chard leaves in a salad and start this container again?

I was going to punt on Sunburn. Lol. Have to agree about the quantity needed for a meal. A great big pile of leaves cooks up to a little pile.

I agree about the spinosad too. Sounds like overkill plus it's known to kill the beneficials that feed on leafminers. So, they end up coming back even worse.

I disagree about the # of plants needed to keep one inundated with Chard. I have 4 plants that keeps putting out more than enough for 3 people for a good couple years. Clip a bunch and the plants are ready to harvest again in the next week.

I would clip all affected leaves(and even some that don't LOOK affected) and maybe transplant them to a spot in the ground or a larger container. You might also want to plant some flowers to attract leafminer, etc. parasitic wasps. Alyssum's a cheap easy one that's long blooming and in the garden centers right now. Research IPM.

Good luck.

Kevin

Here is a link that might be useful: UC IPM

Thanks everyone! It is a shame because I had ~20 heathy chard plants I grew from seed. They are all infested now.
I am growing everything in containers (mostly fabric pots).
Anyone have suggestions for how to put row covers over round pots?
What other plants are vulnerable to these pests? My lettuce and mustard have been left alone; however, I have some spinach and bok choy that just sprouted... They are too tiny for the pests to infest yet. I am also growing radishes and lots of other things from seeds that I recently planted.

I haven't had a garden for 8 years, and this is my first west coast garden (and my largest with the most variety). I am sure will encounter a lot of new pests and problems! So thanks so much for your help, I am sure I will need more :)
I am just so happy to be gardening again.

Thanks,
Robin

They attack all sorts of veggies, but usually the only damage is cosmetic.. the little "mines" that you'll see on leaves. Only thing I really ever noticeably see them attack in my garden is mainly cukes.

Like i said, just get another rather large pot, plant some alyssum, marigolds, maybe some cilantro, dill, parsley. Zinnias, even. And you'll be able to keep quite a few pests in check.

Kevin

Here is a link that might be useful: beneficial attractors