You don't see the enemy but they're there. Leaf miners attack a variety of vegetable crops and landscape specimens, even trees. Adults lay eggs on leaves; the larvae actually enter the leaf tissue and burrow through the inside, eating as they go. It's remarkable that they "know" how to stay within a paper thin leaf, and the mess that tiny miners can make is remarkable too.
When held up to the light, this leaf reveals two tiny miners at work, leaving piles of waste and a ruined leaf
Emphasis on the plural of "miners"
Let me clarify that plural. Leaf miners are not many individuals of one species, but individuals of many species, from several different insect families. Sawflies, moths, beetles, and flies (Liriomyza, Phytomyza) all send juvenile reinforcements to the miner brigade. It doesn't make much difference which clan (order) these insects come from. Most gardeners will never notice the miniscule adults. The eggs they lay are tiny, too, and easily overlooked. As soon as the eggs hatch the larvae enter the leaf and are sheltered inside its cells. Egg-laying adult female insects sometimes bite the leaf tissue and suck juices from it, making scattered small wounds.
Different miners make different, distinctive kinds of tunnels. Some squiggle creatively while others leaves large open patches, or roughly follow the path of leaf veins. Miner damage is a real spoiler to the attractive foliage of some plants and an obvious disaster to edible leafy green crops. Some deciduous and evergreen trees even host leaf or needle miners Dead tissue surrounding the miner's excavation may turn white and papery, have dark patches of miner waste, or look dirty with fungus that takes advantage of the waste.
Control, or not?
I'm happy to tell you that for most ornamentals, leafminers are just an annoyance and not a major threat. Handpicking is the first line of defense for the small vegetable garden and home landscape. With warm weather comes the emergence of the overwintering adults intent on reproduction. This is the time to monitor susceptible plants for eggs (check underneath the leaves), white dots, or the beginnings of a tunnel. Diligently removing and disposing of the first damaged leaves can put a major hurt on the next generations of miners that might otherwise plague your plants later the same year. Weeds and wildflowers can host miners too so remember to remove tunneled leaves on them as well. Check again throughout the growing season for new signs of leaf miners at work. The second line of defense comes easily to gardeners who encourage beneficial insects. Parasitic wasps generally do a very good job of managing (but not eliminating) leaf miners.
Turning the soil in late winter can bury dormant pupae and keep the adults from emerging; floating row covers can keep leaf miners from finding their favorite plants. Crop rotation will move susceptible plants away from miner pupae that dropped the year before.
Insecticide treatment for leaf miners is problematic. The transient adults and larval miners are only briefly exposed to external insecticide sprays, while the parasitic wasps that are your allies are fully vulnerable to the chemicals. Some tree or shrub leaf miners can be treated with carefully timed foliar spray or systemic products; however, trees and shrubs are usually not severely harmed by leaf miners. Well-timed sprays approved for food crops could be helpful in some cases but be sure NOT to use systemic insecticides anywhere near food crops. It seems noteworthy that the sources I found during research recommend only hand picking of damaged leaves and letting beneficial wasps go to work.
Leaf mining insects can cause shockingly extensive looking damage to all kinds of landscape specimens or crops, but most plants are only stressed and not killed by miners. For the home gardener, the best way to deal with miners is through diligent monitoring, using good cultural practices, and maintaining a healthy garden environment that supports beneficial insects and vigorous plants. Chemical treatments have not proven effective enough to warrant their use on what is a mostly a temporary, aesthetic problem.
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John Byers has an excellent page detailing leaf miners and mining insects, here, part of his website, Chemical Ecology of Insects
You may want to read "Attracting Beneficial Insects to the Garden" by Karen Jones