In Riemannian geometry, a theorem originally due to Morse and Littauer states that the exponential map fails to be injective in any neighbourhood of a conjugate vector. Warner also provided an alternative proof of this result by studying some regularity properties and the normal forms of the exponential map. We will discuss a generalisation of these approaches to the sub-Riemannian exponential map and their consequences on the regularity of the sub-Riemannian conjugate locus, as well as on the nature of conjugate points in metric geometry.