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• The spatial collinear restricted four-body problem has been studied. • The peripheral primaries are oblate/prolate spheroid. • The existence and stability of libration points are discussed. • Parametric variation of the regions of possible motions are illustrated. • We unveil the Newton–Raphson basins of convergence, linked with the libration points. In the present work a systematic study has been presented in the context of the existence of libration points, their linear stability, the regions of motion where the test particle can orbit and the domain of basins of convergence linked to libration points in the spatial configuration of the collinear restricted four-body problem with non-spherical primaries (i.e., the primaries are oblate or prolate spheroid). The parametric evolution of the positions of the libration points as function of the oblateness or prolateness parameters of the primaries and the stability of these points in linear sense are illustrated numerically. Moreover, the numerical investigation shows that the only libration points which lie on either of the axes are linearly stable for several combinations of the oblateness parameter and mass parameter whereas the non-collinear libration points are found linearly unstable, consequently unstable in nonlinear sense also, for studied value of mass parameter and oblateness/prolateness parameter. Moreover, the regions of possible motion are also depicted, where the infinitesimal mass is free to orbit, as function of Jacobian constant. In addition, the basins of convergence (BoC) linked to the libration points are illustrated by using the multivariate version of the Newton–Raphson (NR) iterative scheme. [ABSTRACT FROM AUTHOR]