  In this work we explore the propagation of massive scalar fields on some
wormhole backgrounds. On one side, we consider the Bronnikov-Ellis wormhole
solution and wormhole geometries with a non-constant redshift function by
introducing a gravitational mass $M$, which goes over into the Bronnikov-Ellis
wormhole when the gravitational mass parameter vanishes. We employ the
continued fraction method to calculate accurately the quasinormal frequencies
of massive scalar fields, particularly focusing on low values of the angular
number, and we show an anomalous behaviour of the decay rate of the quasinormal
frequencies, for $n \geq \ell$. Also, we show that for a massive scalar field
and $M \neq 0$ the effective potential allows potential wells for some values
of the parameters which support bound states, which are obtained using the
continued fraction method and they are characterized by having only a frequency
of oscillation and they do not decay; however, for the Bronnikov-Ellis wormhole
the effective potential do not support bound states. On the other side, we
consider a wormhole geometry which is an exact solution of $f(R)$ modified
gravity. For this geometry the quasinormal frequencies can be obtained
analytically, being the longest-lived modes the ones with lowest angular number
$\ell$. So, in this wormhole background the anomalous behaviour is avoided.
