The Sturm-Liouville boundary value problem modelling free vibrations of a string resting on a Winkler foundation is studied. The string is assumed to be supported by springs attached to its ends. Along with the conventional elastic springs, those with a negative stiffness are also considered. Nowadays, the latter are exploited in advanced engineering applications. It is shown that the combined effect of the foundation and unusual spring supports may result in non-oscillating eigenfunctions, including trapped modes localised near the ends. In addition, a three-termasymptotic expansion for the higher order eigenvalues is derived from the original transcendental relation. Comparison of asymptotic and numerical results is presented.