I've tried different sites/products and this seems to be split fairly evenly. Windows 7 and Gmail allow you to insert spaces in your password. Hotmail and Twitter do not.
While allowing spaces in a password increases the complexity of a password, it seems like many sites/programs do not allow them. Is there a good reason to allow/disallow spaces?
Here is a quick way to test password strength--use google's own account password API:
https://www.google.com/accounts/RatePassword?Passwd=mypwd
Per your question about whitespace, I have entered a simple password with two characters and one whitespace "t t" . Google gave the password a rating of 3 out of 4. If I do the same password, but remove the whitespace "tt" the rating received is 1 out of 4. By Google's rating standard, including whitespace improves the quality/strength of a password.