A1--
x^2-y^2-z^2
A isolated point
x^2+y^2+z^2
A2--
x^3-y^2-z^2
A2+-
x^3+y^2-z^2
A3--
x^4-y^2-z^2
A3+-
x^4+y^2-z^2
A4--
x^5-y^2-z^2
A4+-
x^5+y^2-z^2
D4--
x^2 y-y^3-z^2
D4+-
x^2 y+y^3-z^2
D5--
x^2 y-y^4-z^2
D5+-
x^2 y+y^4-z^2
D6--
x^2 y-y^5-z^2
D6+-
x^2 y+y^5-z^2
E6--
x^3-y^4-z^2
E6+-
x^3+y^4-z^2
E7
x^3-x y^3-z^2
E8
x^3-y^5-z^2
The above is the complete list of 'simple' singularities.
There are many more which are not simple, such as:
T[3,3,3]
x^3+y^3+z^3
+2 x y z
Not all singularities are isolated. Some surfaces can contain a whole line
of singular points such as:
Whitney Umberella
(cross cap)
x^2 z-y^2
Swallowtail