The PERLAS potential has been successfully used in many studies related to the dynamics of the spiral arms on the equatorial plane of normal (nonbarred) spiral galaxies. In the present work, we extend these studies by investigating the three-dimensional (3D) dynamics of the spiral arms in the same type of potential. We consider a typical open, logarithmic, spiral pattern of pitch angle 25° and we examine the stellar orbits that can support it as the ratio of the masses of the spiral over the disk component (Ms/Md) varies. We indicate the families of 3D periodic orbits that act as the backbone of the spiral structure and we discuss their stability in the models we present. We study further the quasiperiodic and nonperiodic orbits in general that follow spiral-supporting orbits as the Ms/Md ratio increases. We find that a bisymmetric spiral with 25° pitch angle is better supported by orbits in models with . In these cases, a strong spiral pattern is supported between the radial 2:1 and 4:1 resonances, whereas local enhancements of the imposed spirals are encountered in some models between 4:1 and corotation. A characteristic barlike structure is observed in all models at radii smaller than the radius of the 2:1 resonance.