We theoretically study the stability of three dimensional Dirac semimetals against short-range electron-electron interaction and quenched time-reversal symmetric disorder (but excluding mass disorder). First we focus on the clean interacting and the noninteracting dirty Dirac semimetal separately, and show that they support two distinct quantum critical points. Using renormalization group techniques, we find that while interaction driven quantum critical points are Gaussian (mean-field) in nature, describing quantum phase transitions into various broken symmetry phases, the ones controlled by disorder are non-Gaussian, capturing the transition to a metallic phase. We classify such diffusive quantum critical points based on the transformation of disorder vertices under chiral rotation. Our analysis suggests that two distinct quantum critical points are stable in an interacting dirty Dirac semimetal (with chiral symmetric randomness), and a multicritical point results from their interplay. By contrast, the chiral symmetry breaking disorder driven critical point is unstable against weak interactions. Effects of weak disorder on the ordering tendencies in Dirac semimetal are analyzed. The clean interacting critical points, however, satisfy the Harris criterion, and are therefore expected to be unstable against bond disorder. Although our weak coupling analysis is inadequate to establish the ultimate stability of these fixed points in the strong coupling regime, they can still govern crossover behaviors over a large length scale, when either interaction or randomness is sufficiently weak. Scaling behavior of various physical quantities (e.g., spectral gap, specific heat, density of states, conductivity) and associated experimental signatures across various quantum phase transitions are discussed.