We derive a priori estimates for singular differential equations of the form \[ \mathcal{L} φ= P(φ,\nablaφ) + f(φ,\nablaφ)ξ\] where $P$ is a polynomial, $f$ is a sufficiently well-behaved function, and $ξ$ is an irregular distribution such that the equation is subcritical. The differential operator $\mathcal L$ is either a derivative in time, in which case we interpret the equation using rough path theory, or a heat operator, in which case we interpret the equation using regularity structures. Our only assumption on $P$ is that solutions with $ξ=0$ exhibit coercivity. Our estimates are local in space and time, and independent of boundary conditions. One of our main results is an abstract estimate that allows one to pass from a local coercivity property to a global one using scaling, for a large class of equations. This allows us to reduce the problem of deriving a priori estimates to the case when $ξ$ is small.