In the picture, the dart enters through the single 19 area, but the tip goes under the metal separator and sticks in the double 19. What is the correct way to score this?
5.4.2 A scoring dart is counted by reference to the segment into which the point of the dart enters and remains in or touching the face of the dartboard.
Now, what is a segment? There's no explicit definition, but the following rule suggest they'd never expect a situation like in the question to happen:
5.5.7 All the wires forming the doubles and trebles segments, inner and outer centre rings which together form the “Spider”, shall be affixed to the face of the dartboard in such a manner that they lie flat on the face of the dartboard and feature no gaps or splits.
I'd argue that the dart is in the double 19 segment (its tip is fully in the green zone) so it counts as a double.
Now what would happen if it's exactly on the white/green border? There's always a catch-all rule:
5.4.8 The Referee shall act as an umpire in all matters pertaining to these General Playing Rules when conducting a Darts Event and shall, if it is necessary, consult with Scorers and other Officials before announcing any decisions during the course of Match play.
Or perhaps more explicitly:
5.3.9 The Referee will have the final say in all matters relating to starting and finishing.
though this may happen during the 'scoring' phase of a game as well.