  With the developments of the last decade on complete constant mean curvature
1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of such
surfaces are now known. However, most of the known examples have regular ends.
(An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface
has an essential singularity, resp. at most a pole, there.) There are some
known surfaces with irregular ends, but they are all either reducible or of
infinite total curvature. (The surface is reducible if and only if the
monodromy of the secondary Gauss map can be simultaneously diagonalized.) Up to
now there have been no known complete irreducible CMC 1 surfaces in $H^3$ with
finite total curvature and irregular ends. The purpose of this paper is to
construct countably many 1-parameter families of genus zero CMC 1 surfaces with
irregular ends and finite total curvature, which have either dihedral or
Platonic symmetries. For all the examples we produce, we show that they have
finite total curvature and irregular ends. For the examples with dihedral
symmetry and the simplest example with tetrahedral symmetry, we show
irreducibility. Moreover, we construct a genus one CMC 1 surface with four
irregular ends, which is the first known example with positive genus whose ends
are all irregular.
