We give a detailed microlocal study of X-ray transforms over geodesics-like
families of curves with conjugate points of fold type. We show that the normal
operator is the sum of a pseudodifferential operator and a Fourier integral
operator. We compute the principal symbol of both operators and the canonical
relation associated to the Fourier integral operator. In two dimensions, for
the geodesic transform, we show that there is always a cancellation of
singularities to some order, and we give an example where that order is
infinite; therefore the normal operator is not microlocally invertible in that
case. In the case of three dimensions or higher if the canonical relation is a
local canonical graph we show microlocal invertibility of the normal operator.
Several examples are also studied.