Computing a Basis of Holomorphic Differential Forms for a Given Curve I find myself repeatedly having to ask this question, because no one seems to have answered it. I put this up on Math Overflow with a bounty, got a guy who said he would answer it, but—though he helped with other things—never actually got around to answering the question.
Basically, I'm starting to get annoyed.
All I wish to know is this:
Everything is complex numbers. 
Consider a curve $C$ defined by the set of all $\left(x,y\right)\in\mathbb{C}^{2}$ satisfying a formula $f\left(x,y\right)=0$. What is/are the algorithm(s)/formula(e) for a “basis of holomorphic differential forms” for $f\left(x,y\right)$? Let me be clear: I want an algorithm and/or formula I can use by hand to compute bases of holomorphic differential forms using classical, pre-20th century methods for a given curve. Moreover, it would be preferable to keep things in affine complex space as much as possible. 
Examples:
$$4x^{2}y^{2}-\left(x^{2}+y^{2}\right)^{3}=0$$
$$x^{3}-3xy+y^{3}=0$$
$$xy^{3}-x^{4}-2y^{6}x^{2}+2x+1=0$$
$$y^{2}\left(x^{2}+y^{2}\right)-x^{2}+1=0$$
$$xy^{2}-y\cos x+y+\frac{1}{2}=0$$
$$2^{x}y-x^{3}+y^{2}-1=0$$
$$5^{x}y-xy+3^{-y}-1=0$$
Etc. 
To be clear, I've been doing quite a bit of research on the topic, to mixed results. It's clear to me that to compute the basis of differential forms, it suffices to compute a collection of linearly independent expressions (defined by affine equations) whose zero-loci define curves which are adjoint to the curve defined by the formula $f\left(x,y\right)=0$. However, the moment I start asking questions about how to do so, I get bombarded by obfuscating abstractions and generalities—"blow up", "resolution of singularities", "birational equivalence", "sequences of maps", "charts", and the like. Help with wresting the concrete procedures free from the clutches of abstract nonsense would be most appreciated.

If your plane curve is smooth and irreducible, then Theorem 1 on p630 of Brieskorn and Knorrer's "Plane algebraic curves" will answer your question (see also the Corollary on p634).
If your curve isn't smooth then you can't avoid blow ups and resolutions of singularities. Brieskorn and Knorrer's book also has a section explaining how to do this for plane curves.
