No discussion of invasive perennial weeds would be complete without a few words about the 'lowly' dandelion. It's what most folks think of when someone says "weeds" and is no doubt the most widely spread/recognized/hated invasive plant in the U.S. I'd also guess that more money and time are spent trying to control dandelions than any other weed species, noxious or otherwise. Dandelions belong in this part of the overall discussion because, contrary to conventional wisdom, it is a perennial plant. Once established, mature dandelion plants have large tap roots that survive winters as well as most human efforts to control the pesky plants. Almost all dandelion control is done for aesthetic reasons in lawns and turf, since the plants do provide forage for livestock and browsing/grazing wildlife. Since nearly all parts of the dandelion plant are useful as food or medicines, it was probably brought to North America by the earliest European settlers.
Non-Chemical: 1-Mow often to keep the flowers from producing seed and hope your neighbors do it too. 2-dig or pull the plants with one of the many inventions specifically designed for this (you must get nearly all of the root)
3-heavy mulching followed by replanting of the non-target vegetation that is also killed
4-boiling water, agricultural vinegar and flaming torches will kill the tops of the plants making them effective only on young dandelions. Older plants will sprout back from the roots. These methods also kill other plants that are sprayed or burned.
Chemical: 2,4-D was invented for Dandelions. All the various commercially available "lawn herbicides" contain 2,4-D and many also contain small amounts of more serious herbicides such as dicamba or triclopyr to broaden their spectrum of species controlled. Look for "Foundation" or "Triplet" or "Trimec" but be careful not to over-apply these.
New: 1-'Serritor' fungus that kills broad leaf plants only and then "disappears". This may not be available in the US yet.
2-'Fiesta' or 'Iron-X' use chelated iron to give "dandelion, clover and plantain" a lethal dose of iron. This is available at the Sandpoint Co-op as Whitney Farms "Lawn Weed Killer", AND IT SEEMS TO WORK, but only for dandelions. I sprayed a test plot that has grasses and several weed species and it took out the dandelions, but nothing else.This is good news because it won't kill your lawn grasses. However, it remains to be seen if it also killed those deep dandelion tap roots. It comes only in pre-mixed quarts and is expensive on a per acre basis, but for those scattered dandelions it may be the ticket if you want to avoid herbicides.