I know that a surface mapped to a 3-space gives a generic surface with singularity set consists of double points, triple points and branch points (Ofcourse, the singularity set may be empty, such a case gives an embedded surface, i.e standard surface).
A surface in $\mathbb{R}^4$ might be embedded or immersed with singularity set contains only isolated crossing points. I don't have a picture of surfaces in 5 dimension and higher. I have a feeling that any surface in $\mathbb{R}^m$, $m\geq 6$ is a standard surface. This is because the crossing information change is allowed. Could you please help me to get the picture of the surfaces in all dimensions.
A generic surface in 5-space has no singularities. Note, however, that your claims about 3-space and 4-space are false --- you can map a sphere to 3-space so that all points go to the origin $(x, y, z) \mapsto (0,0,0)$ or to a line segment $(x, y, z) \mapsto (0, 0, z)$, or many other possibilities. Only if you assume that the mapping is generic can you make the claims you did (which is why my answer is for generic surfaces).