The critical 2D stochastic heat flow (SHF) arises as the scaling limit of directed polymers in 2+1 dimensions at the critical weak-to-strong disorder transition. It is a random measure expected to exhibit strong intermittency, concentrating its mass on exceptional regions. In this talk, I will discuss two complementary aspects of this phenomenon. First, I will present results on the rapid growth of SHF moments. In particular, we obtain a double-exponential lower bound on moments, matching the growth predicted in the physics literature. I will then turn to the spatial manifestation of intermittency. We show that, while the support of SHF has full dimension at the polynomial scale, it exhibits a precise logarithmic fractal correction. This is based on joint work with Shirshendu Ganguly.