To avoid trival cases, we assume that $f$ is neither a constant nor a finite Blaschke product.
Two celebrated theorems of Frostman say that $f_a(z)$ is actually a Blaschke product for every $|a|<1$ with the possible exception of a set of logarithmic capacity zero and if $w=f(z)$ is not reducing to a finite Blaschke product or a constant, it will assume infinitely often every value of $|w|<1$ except for a possible set of logarithmic capacity zero. According to above two theorems, there are two exceptional sets associating with each inner function $f(z)$:
${\mathcal E}_1(f)=\{w:|w|<1, f(z)$ assumes $w$ at most finitely often$\}$
and
${\mathcal E}_2(f)=\{w:|w|<1, \frac{f(z)-w}{1-{\overline w}f(z)}$ is not a Blaschke product $\}.$
It is known that the points in ${\mathcal E}_2(f)$ are radial limits of $f$ and the inclusion ${\mathcal E}_1(f)\subseteq{\mathcal E}_2(f)$ always holds. However there maybe also exits points in ${\mathbb D}\setminus{\mathcal E}_2(f)$ are radial limits of $f$. Frostman has constructed the following Blaschke product: $$B(z)=\prod_{k=1}^{\infty}\frac{(1-\frac{1}{k^2})-z}{1-(1-\frac{1}{k^2})z}$$ which has the radial limit $0$ at $z=1$. We denote these points by ${\mathcal E}_3(f)$.
questions:
Question 1. ${\mathcal E}_2(f)\setminus{\mathcal E}_1(f)$ is at most a denumerable set?
Question 2. ${\mathcal E}_3(f)$ is at most a finite set?
Question 3. If $a\in{\mathcal E}_2(f)\setminus{\mathcal E}_1(f)$, then the singular measure of the singular part of $f_a(z)$ is discrete(i.e., it consists entirely of point masses)? Where $f_a(z):=\frac{f(z)-a}{1-{\overline a}f(z)} (|a|<1)$.