Multiclass Lighthill-Whitham-Richards traffic models [Benzoni-Gavage and Colombo, Euro. J. Appl. Math., 14 (2003), pp. 587–612; Wong and Wong, Transp. Res. A, 36 (2002), pp. 827–841] give rise to first-order systems of conservation laws that are hyperbolic under usual conditions, so that their associated Cauchy problems are well-posed. Anticipation lengths and reaction times can be incorporated into these models by adding certain conservative second-order terms to these first-order conservation laws. These terms can be diffusive under certain circumstances, thus, in principle, ensuring the stability of the solutions. The purpose of this paper is to analyze the stability of these diffusively corrected models under varying reaction times and anticipation lengths. It is demonstrated that instabilities may develop for high reaction times and short anticipation lengths, and that these instabilities may have controlled frequencies and amplitudes due to their nonlinear nature.