  In this paper the radii of starlikeness of the Jackson and Hahn-Exton
$q$-Bessel functions are considered and for each of them three different
normalization are applied. By applying Euler-Rayleigh inequalities for the
first positive zeros of these functions tight lower and upper bounds for the
radii of starlikeness of these functions are obtained. The Laguerre-P\'olya
class of real entire functions plays an important role in this study. In
particular, we obtain some new bounds for the first positive zero of the
derivative of the classical Bessel function of the first kind.
