  We review the properties of the Morse-Novikov cohomology and compute it for
all known compact complex surfaces with locally conformally K\"ahler metrics.
We present explicit computations for the Inoue surfaces $\mathcal{S}^0$,
$\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally K\"ahler
(and the tamed locally conformally symplectic) forms on $\mathcal{S}^0$. We
prove the nonexistence of LCK metrics with potential and more generally, of
$d_\theta$-exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.
