the singular support
A distribution can be perfectly smooth in some places and rough in others. The singular support pins down exactly where the roughness is — the set of points near which the distribution is not a nice smooth function. Away from the singular support, the distribution agrees with an ordinary infinitely differentiable function; on it, something genuinely non-smooth is happening, like a spike, a jump, or a kink.
Precisely, a point is outside the singular support if the distribution coincides with a smooth function on some neighbourhood of that point. The singular support is the closed set of points where no such smoothing description is possible. For the Dirac delta the only trouble is at the origin, and everywhere else delta is simply the zero function, so its singular support is {0}, the same as its ordinary support. But the two notions can differ sharply: a function with a single corner is smooth except at that corner, so its support might be a whole interval while its singular support is just the corner point.
Singular support matters because the central regularity question for PDEs is where solutions fail to be smooth. Elliptic operators have the celebrated property that they do not create new singularities: the singular support of the solution sits inside the singular support of the source (this is elliptic regularity, or hypoellipticity). For the wave equation, by contrast, singularities travel — the propagation of singularities follows characteristics. Tracking the singular support is how one states these phenomena cleanly.
The Heaviside step H is smooth everywhere except at x = 0, where it jumps. So its support is the whole half-line x >= 0, but its singular support is just the single point {0} — the only place where H is not locally a smooth function.
Support says where it lives; singular support says where it is rough — and they need not coincide.
The singular support is always contained in the support, never larger. A smooth function has empty singular support even if it is nonzero everywhere — being nonzero is not the same as being singular.