  The Lagrange-mesh method is an approximate variational approach having the
form of a mesh calculation because of the use of a Gauss quadrature. Although
this method provides accurate results in many problems with small number of
mesh points, its accuracy can be strongly reduced by the presence of
singularities in the potential term. In this paper, a new regularized
Lagrange-Laguerre mesh, based on \textit{exactly} orthonormal Lagrange
functions, is devised. It is applied to two solvable radial potentials: the
harmonic-oscillator and Coulomb potentials. In spite of the singularities of
the Coulomb and centrifugal potentials, accurate bound-state energies are
obtained for all partial waves. The analysis of these results and a comparison
with other Lagrange-mesh calculations lead to a simple rule to predict in which
cases a singularity does induce or not a significant loss of accuracy in
Lagrange-mesh calculations. In addition, the Lagrange-Laguerre-mesh approach is
applied to the evaluation of phase shifts via integral relations. Small numbers
of mesh points suffice to provide very accurate results.
