Abstract In this work we study a version of the three constant differential-type Oldroyd constitutive relation which allows distinct objective time derivatives for the extra stress and the stretching. We integrate the constitutive equation and determine an equivalent history integral representation for this model for the general class of viscometric motions. For certain choices of the material parameters and initial conditions, we find that this model allows for the development of shear rate discontinuities in the flow domain as a steady viscometric flow is achieved. Correspondingly, we also give evidence that intense shear rate oscillations may occur during the transient period as an impulsively started viscometric flow in a channel tends to a steady state under a constant critical shear stress. This critical shear stress lies in an interval of values for which the material experiences the phenomenon of “flow yielding”. A qualitative comparison with experimental data is made for certain creams and greases. The material instabilities inherent in this constitutive theory for viscometric motions are suggestive of the instabilities that occur in many viscoelastic fluids such as sharkskin patterns, wavy fracture, and spurt flow.