In this paper, we study the self-similar solutions and the time-asymptotic behaviour of solutions for a class of degenerate and singular diffusion equations in the form

$$ u_t=(|(p(u))_x|^{\lambda-2}(p(u))_x)_x,\quad -\infty<x<+\infty,\quad t>0, $$

where $\lambda>2$ is a constant. The existence, uniqueness and regularity for the self-similar solutions are obtained. In particular, the behaviour at two end points is discussed. Based on the monotonicity property of the self-similar solutions and the comparison principle, we also investigate the time convergence of the solution for the Cauchy problem to the corresponding self-similar solution when the initial data have some decay in space variable.