Although where ever possible I am not a user of herbicides, dandelions could be considered an exception.
Even the smallest fragment of root will regenerate and with the production of hundreds of seeds from each plant the potential for regeneration of plants and spread of seedlings is immense.
Dandelions are broad leaved, herbaceous, perenial plants and therefore systemic weedkiller such as round up containing glyphosate or a herbicide containing 2,4-D such as Weed-B-Gon will kill them without damaging grass.
For those who prefer less commercial methods vinegar has been shown to kill dandelions when applied directly onto the leaves. However vinegar is not selective.
If using a commercial weedkiller the active ingredient is translocated from the leaves to the root. Glyphosate is inactivated when it contacts soil. However, the time taken for it to become inactivated can vary from 3 days to over 2 years in Sweden. So the rate of degradation is closely linked with the soil type.
Glyphosate inhibits the action of an enzyme ( 5-enolpyruvylshikimate-3-phosphate synthase) involved in the synthesis of amino acids tyrosine, tryptophan and phenylalanine. It is absorbed through foliage and translocated to growing points it is therefore only effective on actively growing plants.
2,4-dichlorophenoxyacetic acid abbreviated as 2,4-D was one of the first herbicides to be used in the 1940s. It is inexpensive to manufacture and kills many broadleaved plants whilst leaving grasses largely unharmed.
Obviously if dandelions are treated with herbicides they should not be used for food or medicine as they will be contaminated.
Regularly mowing the lawn reduces the height of the dandelions and their leaf area. It is possible to individually remove the plants but it is time consuming.
Dandelion root extracts have been demonstrated to inhibit seed germination and have therefore an allelopathic effect on other plants. Leaf extracts produced much less inhibition.