What are most common methods of …

Artificial methods of vegetative propagation include grafting, layering, cutting. 

a) Grafting involves fusion of tissues of one plant with those of another plant. Grafting is a vegetative method of propagation for apples and roses.The grafted plant consist of a scion and rootstock. Scion is the upper part of grafted plant that forms the shoot where the rootstock is the lower portion of the grafted plant which forms the roots.Scions are selected for good characteristics such as flowers, fruits where rootstocks are selected as the plant benefit through the efficient intake of nutrients from soil as well for disease, pest and drought resistence.In root stock grafting scion of desired plant is inserted to a rootstock by making an inward cut in the rootstock and sealing the graft area.

Layering: This involves bending of a young stem towards the ground and burying it under the soil for development of roots . After a period of time, as the roots develop, the bent stem is cut off from the parent plant. This acts as a new plant. e.g. Jasmine.   

Cutting: It involves planting a young cutting of the stem with buds into moist soil. This develops roots which absorb nutrition from the soil and help in the growth of new plant. e.g. Bougainvellia. Cutting: It involves planting a young cutting of the stem with buds into moist soil. This develops roots which absorb nutrition from the soil and help in the growth of new plant. e.g. Bougainvellia.