A linear morphological stability analysis of a planar solid-liquid phase interface describing the solidification processes of a binary melt with convection is carried out. The developed theory includes conductive and convective heat and mass transfer mechanisms near the phase interface and generalizes previously known theories of morphological stability. The amplification rate as a function of wavenumber of perturbations and neutral stability curve that divides the stability/instability parametric domains are obtained. It is shown that these domains are highly dependent of convection intensity, which represents a stabilizing factor for solidification processes. A criterion of concentration supercooling in the steady-state solidification conditions with convection is found. The obtained dispersion relation and neutral stability curve define various crystallization scenarios such as (i) morphological instability and concentration supercooling appearing to the formation of a two-phase mushy layer, (ii) morphological stability and concentration supercooling leading to the existence of a slurry layer, (iii) morphological stability without concentration supercooling when the planar solidification front is stable, and (iv) morphological instability without concentration supercooling forming the mesoscopically rough phase interface.