The Multiscale Finite Element Method (MsFEM) is a finite element (FE) approach that allows to solve partial differential equations (PDEs) with highly oscillatory coefficients on a coarse mesh, i.e. a mesh with elements of size much larger than the characteristic scale of the heterogeneities. To do so, MsFEMs use pre-computed basis functions, adapted to the differential operator, thereby taking into account the small scales of the problem. We consider here multiscale advection-diffusion problems in the convection-dominated regime (see [R. Biezemans, C. Le Bris, F. Legoll and A. Lozinski, CMAME 2025]). When the PDE contains dominating advection terms, naive FE approximations lead to spurious oscillations, even in the absence of oscillatory coefficients. Stabilization techniques (such as SUPG) are to be adopted. In the multiscale context considered here, we discuss different ways to define the MsFEM basis functions, and how to combine the approach with stabilization-type methods. In particular, we show that methods using suitable bubble functions and Crouzeix-Raviart type boundary conditions for the local problems turn out to be very effective. Joint work with Rutger Biezemans, Claude Le Bris (ENPC and Inria) and Alexei Lozinski (Univ. Franche-Comte).