  A usual assumption in the so-called {\it de Broglie - Bohm} approach to
quantum dynamics is that the quantum trajectories subject to typical `guiding'
wavefunctions turn to be quite irregular, i.e. {\it chaotic} (in the dynamical
systems' sense). In the present paper, we consider mainly cases in which the
quantum trajectories are {\it ordered}, i.e. they have zero Lyapunov
characteristic numbers. We use perturbative methods to establish the existence
of such trajectories from a theoretical point of view, while we analyze their
properties via numerical experiments. Using a 2D harmonic oscillator system, we
first establish conditions under which a trajectory can be shown to avoid close
encounters with a moving nodal point, thus avoiding the source of chaos in this
system. We then consider series expansions for trajectories both in the
interior and the exterior of the domain covered by nodal lines, probing the
domain of convergence as well as how successful the series are in comparison
with numerical computations or regular trajectories. We then examine a
H\'{e}non - Heiles system possessing regular trajectories, thus generalizing
previous results. Finally, we explore a key issue of physical interest in the
context of the de Broglie - Bohm formalism, namely the influence of order in
the so-called {\it quantum relaxation} effect. We show that the existence of
regular trajectories poses restrictions to the quantum relaxation process, and
we give examples in which the relaxation is suppressed even when we consider
initial ensembles of only chaotic trajectories, provided, however, that the
system as a whole is characterized by a certain degree of order.
