  We refine the null alignment classification of the Weyl tensor of a
five-dimensional spacetime. The paper focusses on the algebraically special
alignment types {\bf {N}}, {\bf {III}}, {\bf {II}} and {\bf {D}}, while types
{\bf {I}} and {\bf {G}} are briefly discussed. A first refinement is provided
by the notion of spin type of the components of highest boost weight. Second,
we analyze the Segre types of the Weyl operator acting on bivector space and
examine the intersection with the spin type classification. We present a full
treatment for types {\bf {N}} and {\bf {III}}, and illustrate the
classification from different viewpoints (Segre type, rank, spin type) for
types {\bf {II}} and {\bf {D}}, paying particular attention to possible
nilpotence, which is a new feature of higher dimensions. We also point out
other essential differences with the four-dimensional case. In passing, we
exemplify the refined classification by mentioning the special subtypes
associated to certain important spacetimes, such as Myers-Perry black holes,
black strings, Robinson-Trautman spacetimes, and purely electric/magnetic type
{\bf {D}} spacetimes.
