How do you find vertical, horizontal and oblique asymptotes for f(x)= x^(1/3)?

This function does not have any asymptotes. It has some nice rotational symmetry, because it is an "odd" function. It is the inverse of $f \left(x\right) = {x}^{3}$ which means that its graph would be a reflection across the line y = x. If you wish to investigate asymptotes for functions, perhaps take a look at "Rational" functions.