Fully understanding the mechanism of pore-scale immiscible displacement dominated by capillary forces, especially local instabilities and their influence on flow patterns, is essential for various industrial and environmental applications such as enhanced oil recovery, ${\mathrm{CO}}_{\mathrm{2}}$ geo-sequestration and remediation of contaminated aquifers. It is well known that such immiscible displacement is extremely sensitive to the fluid properties and pore structure, especially the wetting properties of the porous medium which affect not only local interfacial instabilities at the micro-scale, but also displacement patterns at the macro-scale. In this review, local interfacial instabilities under three typical wetting conditions, namely Haines jump events during weakly-wetting drainage, snap-off events during strongly-wetting imbibition, and the co-existence of concave and convex interfaces under intermediate-wet condition, are reviewed to help understand the microscale physics and macroscopic consequences resulting in natural porous media.