Ferroelasticity is a prominent material property analogous to ferroelectricity and ferromagnetism, but its characteristic spontaneous structural polarization has remained less studied and poorly understood. Here, we use a high-throughput computation approach in conjunction with first-principles calculations to identify 65 (M=transition metal, X=nonmetal) monolayers exhibiting in-plane ferroelasticity out of 166 stable tetragonal monolayers. Molecular orbital theory analysis reveals that ferroelastic distortion arises when M-d/X-p and M-d/M-d couplings are both sufficiently weak. We have developed a physically interpretable one-dimensional descriptor that correctly predicts 89% of ferroelastics or nonferroelastics among the examined MX monolayers. Moreover, we find eleven MX compounds that exhibit strongly coupled ferroelasticity and magnetism driven by strain-controlled magnetocrystalline anisotropy, raising the prospects of developing 2D ferroelasticity-based multiferroics.