History: Flower offerings are of Buddhist origin; originally with 2 flowers left of the altar, 2 candles to the right and a censer in the middle; later then only one each. This is still done today, for example, for 利休記.
They can either be placed on the floor, hung on the wall (or 床柱) or hung from the ceiling (usually boats or half-moons).
They can also be categorised according to material, giving us a 真・行・草 hierarchy. However, the shape can interfere with the categorisation, independent of the material.
真: metal things, celadon, porcelain.
行: All boat shapes (even bamboo), glazed Japanese wares.
草: Unglazed Japanese wares, 楽; gourds, baskets and bamboo. Also, glass, and みたて wares.
Of course, there's also a 風炉・炉 divide: Baskets and gourds are for summer; except for lacquered Chinese baskets, which can be used all year.
薄板
真: 真塗, 矢筈
行: 蛤歯, 真塗
草: 蛤歯, unlacquered wood (Baskets don't need a board)
Bamboo 花入
In 1590, 利休 made 3 famous bamboo 花入: 尺八, よなが, 音常時
The latter Hideyoshi didn't like, but rikyu sent it to Oribe
In 1660, Onjoji vase was rated Japan's best vase.
Copper 花入
kyo-zutsu, the shape of a sutra container.
鶴の一声 or 鶴の端 (Crane's voice) was a vase that 利休 was really fond of.