After providing celebrities and big brands with customizable vanity Google+ URL's the company is now slowly rolling out the feature to other established accounts. I was lucky to get one with my full-name "Deepesh Agarwal" as changing this still requires manual review from Google moderation team. To get these custom, short and easy-to-remember web addresses that links directly to your profile or G+ page, Google requires users to meet some eligibility requirements which are Ten or more followers, an account which is at-least 30 days old and a profile photo.

There is no way to signup or get custom URL from user's end as of now, if a user meets the minimum requirements set by Google they should be getting an email as this gets rolled out to more and more accounts.