In Milnor's book "Hypersurface singularities" He discusses shortly knots that arrive as Milnor spheres of algebraic curves, i.e knots that are the intersection of a $3$ sphere around a singular point of a complex algebraic curve in $\mathbb{C}^{2}$.
I am looking for references on these types of knots. Specifically, I am interested in their properties and in examples of knots that are not this type, together with proof. Preferably statements about knot invariants of these knots.
Knots and, more generally, links appearing as in links of algebraic hypersurface singularities in ${\mathbb C}^3$ are called algebraic. They are rare among all (tame) knots and links in $S^3$. They can be described as obtained using torus knots and links via certain constructions (cabling). For instance, they can never be hyperbolic and, hence, the simplest example of a non-algebraic knot is the figure eight knot.
You can find a detailed study of algebraic links (with proofs of the above statements) in the book
D.Eisenbud, W.Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Princeton University Press, 1986.