There are some people out there that are having success with Tenacity. Ethofumasate (Prograss) is a selective herbicide that targets poa annua. Velocity is another one, but that one might be restricted to professionals only, and it can cause temporary harm to KBG. A good anti-poa annua strategy involves regular preemergent applications, especially in late summer, and late winter/early spring. Controlled round-up application will obviously kill poa annua, but if you don't have a spreading grass like KBG, those areas will remain bare, but you can always plant some grass in pots to fill in the bare areas since you will have a preemergent barrier down.

I used Round-Up to kill the lawn and reseeded in August. I had to pull and reseed some spots the second year. Year 3 my tall fescue has taken over and I have a good lawn. Grass seed in the fall and use peremergence in the spring. Look for sources of the PoaAnnua in fence rows and edges of walks and driveway. I hate it worse than crab grass.

But here's the rub, poa annua, is a winter annual (there are actually many varieties, some are perennials, just to make your life that much better). So what that means is the seeds sprout in fall, and the plant lies dormant in a vegetative state over winter, not dying like bpgreen stated. Then come spring it produces many seed heads, and can do so at very low mowing heights. (Seeds in the soil can germinate in spring as well.) It's these seed heads that most people find so offensive, plus it grows faster than most turf and is lighter green as well, so it sticks out like a sore thumb even when it isn't producing seed. It usually dies in the summer from the heat, but at that point it has put down thousands of seeds ready to sprout when soil temps are lower in the fall. There are a few selective herbicides out there that will kill poa annua, like bispyribac-sodium (Velocity) and ethofumesate (Prograss), but they are expensive, and not easy to use. Usually a few years of preemergents in the late summer and early spring will greatly diminish poa annua in your lawn, but that also means you can't overseed because of the preemergents.

Poa annua germinates mainly in the fall and there are gazillions of seeds lying around waiting to germinate then. So the best way to get rid of it is to use pre-emergent in early fall. Early fall is also the time to plant cool season grass, so if you also want to overseed, you can't use normal pre-emergent then. Some people use pre-emergent in the fall to take care of the poa annua and then seed in the spring to get new grass. Spring seeded grass has its own issues (can't use pre-emergent to take care of summer crabgrass, summer heat overwhelms inadequate root systems). If you have lots of shade, it's always going to be a problem to get grass to grow. Have you considered alternative groundcovers? Shade garden plants (ferns, hostas, etc.)? Mulch and shrubs?