Drupal does not have a hard standard for the use of single quotes vs. double quotes. Where possible, keep consistency within each module, and respect the personal style of other developers.

With that caveat in mind: single quote strings are known to be faster because the parser doesn't have to look for in-line variables. Their use is recommended except in two cases:

1. In-line variable usage, e.g. "<h2>$header</h2>".
2. Translated strings where one can avoid escaping single quotes by enclosing the string in double quotes. One such string would be "He's a good person." It would be 'He\'s a good person.' with single quotes. Such escaping may not be handled properly by .pot file generators for text translation, and it's also somewhat awkward to read.