I asked myself the following question while preparing a course on power series for 2nd year students. Let $F$ be the set of power series with convergence radius equal to $1$. What subsets $S$ of the unit circle $C$ can be realised as $$ S:=\{x \in C: f\text{ diverges in }x\} $$ for $f \in F$? Any finite subset (and possibly any countable subset) of $C$ can be realised that way. Who knows more on this?

$\begingroup$ If your countable set is closed, then a suitable series is relatively easy to construct. $\endgroup$– S. Carnahan ♦Dec 14 '10 at 15:56

3$\begingroup$ Until Andres return you can look at the paper below (can be found by google search) which has a lot of information. In particular any $G_\delta$ set and any $F_\sigma$ sets of logarithmic measure zero is a set of divergence. Erdös, Paul; Herzog, Fritz; Piranian, George, Sets of divergence of Taylor series and of trigonometric series. Math. Scand. 2, (1954). 262–266. $\endgroup$– Gideon SchechtmanDec 14 '10 at 16:27

1$\begingroup$ Note the mention of Fourier series: If power series $\sum a_n z^n$ has radius of convergence 1, write $z=e^{i\theta}$ to get a Fourier series, whose set of divergence (in the real line) you are interested in. $\endgroup$– Gerald EdgarDec 14 '10 at 17:26

$\begingroup$ @Gerald: The issue is that many of the trigonometric series that appear here are not Fourier series. $\endgroup$– Andrés E. CaicedoDec 14 '10 at 17:48

4$\begingroup$ $\sum z^n/n^2$ converges on the whole circle. $\endgroup$– Gerald EdgarDec 15 '10 at 18:16
[Edit (Jan.12/12): I've recently been made aware of additional results. I have added an update at the end.]
Hi Piotr,
As far as I know, the question is still open, and not much seems known beyond the classic results of George Piranian and Fritz Herzog. They are contained in two joint papers, available at the publications page of Piranian's website (although the quality of the scans is not optimal),
 "Sets of convergence of Taylor series. I." Duke Math. J. 16, (1949), 529534, and
 "Sets of convergence of Taylor series. II." Duke Math. J. 20, (1953), 4154.
Suppose $f(z)=\sum_na_nz^n$, and let $f_n(z)=\sum_{k\le n}a_kz^k$. The Cauchy convergence criterion says that $f(z)$ converges iff $$ \forall n\exists N\forall m,k\ge N f_m(z)f_k(z)\le 1/n. $$ This shows clearly that the set of $z$ on the unit circle where $f$ converges is a countable intersection (indexed by $n$) of a countable union (indexed by $N$) of closed sets (namely, the intersection over $m,k\ge N$ of the sets $\{z\in C\midf_m(z)f_k(z)\le 1/n\}$).
The conjecture is that any subset of $C$ that is a countable intersection of a countable union of closed sets is the set of convergence of some $f$. [Edit (Jan.12/12): This is not the case. See the update at the end.] These things are called $F_{\sigma\delta}$, though I've never quite liked the name.
I have been interested in the problem (on and off) for a bit, so I have asked around what is known. Both Donald Sarason and Steve Krantz agreed that the state of the art seems to be whatever you find in the two papers listed above (Herzog and Piranian have a few papers on closely related topics, some may also be relevant). Alekos Kechris also pointed out that Ted Kaczynski, who was Piranian's student, had worked on related problems. There is a family of questions in the neighborhood of this one, that descriptive set theorists find interesting, so Alekos and other descriptive set theorists had followed some of this work for a little while.
In his thesis, Kaczynski proved that if we look at analytic functions $f$ the upper half plane, then the $F_{\sigma\delta}$ subsets of the real axis are precisely the sets $A$ for which there is such an $f$ so that $A$ is the set of points $p$ on the real axis such that there is an arc $\gamma$ ending at $p$ such that the limit of $f(z)$ as $z$ approaches $p$ along $\gamma$ exists. In his papers he finds several extensions of this result. It seems to me that his techniques are highly relevant to the problem you are asking.
I emailed Piranian a few years ago, but sadly he never replied to my email, it would have been nice to meet him.
The results in the two papers above are as follows: In the first one, Herzog and Piranian trace the history of the problem (for example, Mazurkiewicz had shown that any closed subset of $C$ is the set of convergence of some $f$). They prove that every $F_\sigma$ set is a set of convergence. Their technique is nice (it has a recursion theoretic flavor to it, at least for me, ensuring convergence in some regions and divergence in some others; it is akin to a priority argument), and it looks to me like the "right" kind of approach to this problem. A very simple version of their argument can be found in Rudin's analysis book, where it is shown that $$ \sum_n \frac{z^n}n $$ converges for all $z\in C$ other than $z=1$. The general argument requires some easy trigonometric estimates (that they establish in sections 2,3).
In the second paper, they build some examples of $F_{\sigma\delta}$ sets that are not $F_\sigma$ and yet are sets of convergence. These constructions are more ad hoc, at least to my untrained eyes, they are certainly much more involved than the results in the first paper. There are two proofs given there: The first elaborates the techniques of the paper by ErdösHerzogPiranian mentioned by Gideon in a comment to the question, and so the functions that are obtained are injective in the unit disk. Granting injectivity is one of the main reasons for the difficulty of the approach. The second proof elaborates the approach of the first paper (so injectivity is not ensured here), and uses ideas of Fejér (as mentioned by Theo in a comment below). The first argument allows one to deal with questions of uniform convergence; the second seems more general but does not grant this control. As the authors point out (and thanks to Theo for mentioning this in his comment) Fejér's techniques suffice to show that there are non$F_\sigma$ sets of convergence. Unfortunately, I am not convinced these techniques alone suffice to establish the full characterization.
Hope this helps! I am very curious to see if there are additional results I've missed.
Update (Jan.12/12): Essentially the same question was asked on the sister site. Dave Renfro mentioned a reference I wasn't aware of: Thomas W. Körner, "The behavior of power series on their circle of convergence", in Banach Spaces, Harmonic Analysis, and Probability Theory, Springer Lecture Notes in Mathematics #995, SpringerVerlag, 1983, 5694.
This is a nice survey, its only problem is that it lists no references. Körner proves a result there that goes beyond what I discuss above and shows that the question of what sets are sets of convergence is not purely topological.
Theorem (Körner). There is a $G_\delta$ subset of $C$ that is not a set of convergence.
(This is theorem 6.2 in the paper.) The theorem builds on a result of Marcinkiewicz, who showed that if $a_n\to0$ as $n\to\infty$, then $\sum_{N}^N a_n \exp(int)$ cannot diverge boundedly everywhere as $N\to\infty$. This can be refined to show that, under the same assumptions, if $\sum_{N}^N a_n \exp(int)$ is bounded for all $t$ in some interval $I$, then there is a subinterval $J\subseteq I$ where $\lim_{N\to\infty}\sum_{N}^N a_n \exp(int)$ converges almost everywhere.
The construction makes use of (Cantorlike) sets of positive measure, and in particular leaves the following open:
Is every $G_{\delta\sigma}$ set of Lebesgue measure zero the complement of a set of convergence?
(Recall that Katznelson and Kahane have shown that every set of Lebesgue measure zero has a Fourier series diverging on it, and possibly elsewhere, see also this question.)
I contacted Körner, who indicated he didn't know of additional results. In particular, he mentioned the question above. He also suggested looking into some recent results of Matheron (in particular, the MatheronZelený 2005 paper in Fundamenta), which may suggest descriptive set theoretic approaches to the characterization problem.

$\begingroup$ I think it's worth pointing out that the main effort of the second paper is to provide examples of schlicht (= injective and holomorphic in the open unit disk) Taylor series whose points of convergence not $F_{\sigma}$ on the unit circle. They also mention that the work of Féjer is sufficient to produce examples of non$F_{\sigma}$sets of divergence (I believe that this is tightly connected with Gerald Edgar's comment on Fourier series). $\endgroup$ Dec 14 '10 at 18:17

1$\begingroup$ @Andres Caicedo: That's a very thorough answer! $\endgroup$ Dec 14 '10 at 18:35

2$\begingroup$ I'll just point out that the two papers Andres is referring to are both available at Project Euclid: dx.doi.org/10.1215/S0012709449016476 and dx.doi.org/10.1215/S0012709453020055 $\endgroup$ Dec 15 '10 at 2:00

1$\begingroup$ The question came up on SE again: math.stackexchange.com/q/288765 user mrf refers to Lukašenko S. Ju., Sets of divergence and nonsummability for trigonometric series, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1978, no. 2, 65–70 for the result that there is a $G_\delta$set which is not a set of convergence. $\endgroup$– MartinJan 28 '13 at 10:12

1$\begingroup$ @jaRedDRedmp Hello. I believe that measuretheoretic considerations will play a key role in any characterization. They certainly seem to appear prominently in "recent" results. Körner's survey mentions some of them and includes some speculations of his own along these lines. $\endgroup$ Apr 19 at 19:41