Celestial CFTs are putative 2D holographic dual theories for quantum gravity in 4D asymptotically flat spacetimes. Celestial operators with integer conformal dimensions organize the so-called conformally soft sector of celestial CFTs, providing 2D currents for the theory. In this talk, I will discuss the presence of logarithmic CFT structures in the soft sector of celestial CFT associated with supertranslation symmetries.