In the framework of harmonic and Clifford analysis, specific distributions in Euclidean space of arbitrary dimension, which are of particular importance for theoretical physics, are once more thoroughly studied. Indeed, actions involving spherical coordinates, such as the radial derivative and multiplication and division by the radial distance, only make sense when considering so-called signumdistributions, that is, bounded linear functionals on a space of test functions showing a singularity at the origin. Introducing these signumdistributions, the actions of a whole series of scalar and vectorial differential operators on the distributions under consideration, lead to a number of surprising results, as illustrated by some examples in three-dimensional mathematical physics.