Approaching a branch point along different paths

Question

There's a very nice characterization of the three main types of isolated singularities of an analytic function $f(z)$ that takes oriented curves $\gamma$ that terminate at the singularity and considers the value of $f(\gamma)$ along those curves:

1. An isolated singularity is removable iff $f(\gamma)$ approaches the same limiting value for any curve $\gamma$.
2. An isolated singularity is a pole iff $|f(\gamma)| \to \infty$ for any curve $\gamma$.
3. An isolated singularity is essential iff for any limiting value $u \in \mathbb{C}$, there exists a curve $\gamma$ such that $f(\gamma) \to u$.

These are the only possibilities for an isolated singularity.

Similarly to the case of a removable singularity, for any algebraic branch point, $f(z)$ approaches the same limiting value along any curve $\gamma$ that terminates at that point (e.g. if $f(z) = z^{1/n},\ n \in \mathbb{N}$, then $f(\gamma) \to 0$ along any path $\gamma$ that terminates at $0$). Are there similar characterizations for transcendental and logarithmic branch points? It seems hard to find any:

1. The function $f(z) = \exp\left[z^{-1/2}\right]$ has a non-logarithmic transcendental branch point at $z=0$. If we approach the branch point along the negative real axis, the value of $f(\gamma)$ circles around the complex unit circle faster and faster without ever approaching any fixed value (including $\infty$). If we approach it from any other direction, (I think) $f(\gamma)$ approaches either $0$ or $\infty$, depending on which sheet we approach from. Is there any general statement we can make about the set of possible limit points?

2. The function $f(z) = \ln(x)$ has a logarithmic branch point at $z = 0$. As we approach it along any curve $\gamma$, we have that $|f(\gamma)| \to \infty$, as with a pole. On the other hand, the dilogarithm function $\mathrm{Li}_2(z)$ has a logarithmic branch point at $z = 1$, but I believe that it is continuous at this singularity, in the sense that $f(\gamma) \to \pi^2/6$ for any curve $\gamma$ that terminates at $z = 1$.

It seems that in general, a much wider variety of limiting behaviors is possible as one approaches transcendental and logarithmic branch points along different paths.