  The character of motion for the three-dimensional circular restricted
three-body problem with oblate primaries is investigated. The orbits of the
test particle are classified into four types: non-escaping regular orbits
around the primaries, trapped chaotic (or sticky) orbits, escaping orbits that
pass over the Lagrange saddle points $L_2$ and $L_3$, and orbits that lead the
test particle to collide with one of the primary bodies. We numerically explore
the motion of the test particle by presenting color-coded diagrams, where the
initial conditions are mapped to the orbit type and studied as a function of
the total orbital energy, the initial value of the $z$-coordinate and the
oblateness coefficient. The fraction of the collision orbits, measured on the
color-coded diagrams, show an algebraic dependence on the oblateness
coefficient, which can be derived by simple semi-theoretical arguments.
