Which degree sequences are planar graphical? The Erdős–Gallai theorem characterizes which degree sequences are graphical (i.e. realizable by a simple graph).
There has been some work on which degree sequences are planar graphical (i.e. realizable by a simple planar graph).  See, for example, On Planar Graphical Degree Sequences by Schmeichel and Hakimi (1977).
What is currently known about which degree sequences are planar graphical?
 A: It is always difficult to say what is "currently known," but at least around
2008, the paper

"A Characterization of the degree sequences of 2-trees."
  Prosenjit Bose, Vida Dujmovi, Danny Krizanc, Stefan Langerman, Pat Morin, David R. Wood, Stefanie Wuhrer.
  Journal of Graph Theory
  Volume 58, Issue 3, pages 191–209, July 2008.
  (journal link)

gives a pretty definitive summary of the state of the art:

That a sequence of $n$ positive integers is the degree sequence of a tree if and only
  if it sums to $2n − 2$ is a folklore result. Other graph
  families with known degree sequence characterizations
  include split graphs [14, 20], $C_4$-minor free graphs [21],
  unicyclic graphs [1], cacti graphs [15], Halin graphs [2],
  and edge-maximal outerplanar graphs [18]. The most
  investigated class of graphs is that of planar graphs.
  Despite the eﬀort, no characterization of the degree
  sequences of planar graphs is known, even for edge-maximal planar graphs. 
  Partial results are obtained in
  [5, 13, 11, 16].

And, obviously, that paper characterizes 2-trees.
