The title explains it all.

I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are characterized as those isolated double points that admit a resolution that is given by stepwise blowup of isolated double points.

My question now is whether other type of multiplicity two points exist, i.e. nonrational double points? If so, can they occur on a surface embedded in $\mathbb{P}^3$?

Due to Reids characterization i would say these are isolated double points such that somewhere in the blowup process either a singular point of multiplicity $>2$ or a singular curve appears. Both would seem weird to me, but perhaps my intuition is not correct in this case. Any comments on this aspect are highly appreciated!