What type of alcohol is the crotonyl alcohol?

Crotonyl alcohol has the structure $\mathrm{CH}_3-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}_2-\mathrm{OH}$, where the hydroxyl group is bonded to a saturated carbon atom adjacent to a carbon-carbon double bond. This defines it as an allylic alcohol.

Among the options, allylic alcohol describes an alcohol with the $-\mathrm{OH}$ group on an $\mathrm{sp}^{3}$ carbon adjacent to a double bond, benzylic alcohol requires an aromatic ring, vinylic alcohol has the $\mathrm{-OH}$ directly attached to an $\mathrm{sp}^{2}$ carbon of the double bond, and polyhydric alcohol contains multiple hydroxyl groups. Only allylic alcohol matches the structure.

The classification is therefore $\boxed{A}$.