Purpose. The work is devoted to the study of horizontal oscillation and the assessment of the motion stability of a single wheelset with independent wheel rotation, and to the comparison of stability indicators of the typical wheelset and the wheelset with independent wheel rotation. This is connected with the necessity to increase traffic speed of rolling stock, improve road safety and comfort of passengers. Methodology. To achieve this purpose we used the methods of mathematical simulation of railway rolling stock dynamics, as well as the linear algebra methods to assess the stability of solutions of the linear homogeneous differential equations. Findings. To solve the set task the design model of a single wheelset with independent wheel rotation was created. The wheelset is not a single solid body; each of the wheelset axles has a surplus degree of freedom. Thus, we obtained the system with 4 degrees of freedom. The design model allowed to obtain the system of linear homogeneous differential equations describing the oscillations of the represented wheelset in a horizontal plane on a straight track section. On the basis of the computer modeling were calculated the eigenvalues of the differential equation system coefficients and the asymptotic stability analysis of the wheelset motion with independent wheel rotation. The increment and the frequency of fluctuations were compared with similar indicators for the standard wheelset. The authors also discussed non-oscillatory forms of the wheelset motion and the issues of wheelset self-centering on the track. Originality. The result of the work is the mathematical model of the sinuous movement of a single wheelset, in two-dimensional formulation, with independent wheel rotation and the estimate of the dynamic indices during its motion on a straight track section without any irregularities. There were also proposed the ways to ensure the self-centering on the track of the wheelset with independent wheel rotation. Practical value. The developed mathematical model of the single wheelset motion with independent wheel rotation can be used to create the advanced designs of railway running gear of cars.