I'm applying to several math PhD programs for the next fall term, and they all require a statement of purpose of 1-2 pages. However, none of them seem to give concrete spacing requirements. My statement is too long to fit under double spacing, but meets the requirement under single spacing. I've looked up this question but I've gotten mixed messages.
It probably doesn't matter, although you should review the application instructions carefully just to make sure they don't specify this.
My impression is that admissions committees don't care much about issues like this; certainly I don't care when I serve on these committees. In particular, I have no idea how many applicants have used single or double spacing, since I've never paid any attention to this issue.
The purpose of the length restriction is twofold: to keep the committee from having to read unpleasantly long documents, and to keep applicants from feeling they have to write a lot. The precise cut-off is not so important, which is why it is usually not specified carefully (for example, by a word count or detailed formatting instructions).
You may annoy the committee if it looks like you are trying to cram as much as possible into your statement of purpose. If you use a 10pt font and tiny margins as well as single spacing, it will look like you are gaming the system and trying to write a longer statement of purpose than anyone else. That's not the impression you want anyone to have of your statement. However, I don't think single spacing is problematic in itself. Just don't go overboard with space-saving tricks. (If you use an 11pt font and standard LaTeX margins, you'll be fine.)