  We prove that a compact Vaisman manifold $(M, J)$ cannot admit some type of
special Hermitian metrics, such as special $k$-Gauduchon metrics, $p$-K\"ahler
forms, Hermitian-symplectic or strongly Gauduchon metrics compatible to the
same complex structure $J$. In particular, it cannot admit pluriclosed or
balanced metrics. We also investigate the interplay between locally conformally
symplectic forms taming the complex structure $J$ and special Hermitian
structures.
