We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of
Legendrian characteristic classes such that any Legendrian Thom polynomial has nonnegative coefficients when expanded in these
bases. The method uses a suitable Lagrange Grassmann bundle on the product of projective spaces. This is an extension of a nonnegativity result for Lagrangian Thom polynomials obtained by the authors previously. For a fixed specialization, other special-
izations of the parameter lead to upper bounds for the coefficients of the given basis. One gets also upper bounds of the coefficients
from the positivity of classical Thom polynomials (of singularities of mappings), obtained previously by the last two authors.