We proved Hölder regularity for the particle trajectories of an interacting particle system in one dimension. The particle velocities were given by the nonlocal and singular interactions with the other particles. Particle collisions occur in finite time. Prior to collisions the particle velocities become unbounded, and thus the trajectories fail to be of class $C^1$. Our Hölder-regularity result supplements earlier studies on the well-posedness of the particle system which implies only continuity of the trajectories. Moreover, it extends and unifies several of the previously obtained estimates on the trajectories. Our proof method relied on standard ODE techniques: we transformed the system into different variables to expose and exploit the hidden monotonicity properties.