  We present a general classification of one-soliton solutions as well as novel
families of rogue-wave solutions for $F=1$ spinor Bose-Einstein condensates
(BECs). These solutions are obtained from the inverse scattering transform for
a focusing matrix nonlinear Schr\"odinger equation which models condensates in
the case of attractive mean field interactions and ferromagnetic spin-exchange
interactions. In particular, we show that, when no background is present, all
one-soliton solutions are reducible via unitary transformations to a
combination of oppositely-polarized solitonic solutions of single-component
BECs. On the other hand, we show that, when a non-zero background is present,
not all matrix one-soliton solutions are reducible to a simple combination of
scalar solutions. We show that some solitons are topological ones and others
are dark-bright solitons. Finally, by taking suitable limits of all the
solutions on a non-zero background we also obtain three families of rogue-wave
(i.e., rational) solutions, two of which are novel to the best of our
knowledge.
