Updated Monday, August 27, 2012
Over last few days, Google has invited early Google Profile users to claim a vanity URL and left those not invited wondering how this all might work.
We have received questions from "Do you know the link where one can claim a vanity URL?"; to, "can we call someone?". This evening, I received my notification to claim my Google+ URL: "www.google.com/+BirgitPauli-Haack".
These are the steps.
Step 1: Email notification from Google+
The schema of a Google+ Vanity URL comprises of www.google.com and then /+Yourname. Once you click on the red button on the bottom of the notification you will get to the next screen where you can decide if the suggested name is what you desire or not. You will get a link to claim a different name, if the combination of first name and last name doesn't work for you. When you try to come up with your variation, make sure it follows the convention for folder names. Only alphanumeric characters and only a few punctuation choices.

Step 2: A click on the red button "Claim URL" opens your Google+ Profile page, with an overlay windwow:

If you haven't yet, take a good look at the URL suggested by Google. On the bottom right, next to the buttons is a small link: "Don't like these URLS? Request a different one."
Here is a screen shot of the detour, thanks to Danny Sullivan from Search Engine Land.

I was quite happy with the choice given /+BirgitPauli-Haack, but you might have a different idea than Google. Try it out and see if it's available.
To successfully claim the URL make sure you check the box and agree to the Terms of Service and click on the blue button. You can read the Terms of Service in advance and if you suffer from insomnia and one page is not enough, they offer you three additional policy documents as a cure. (Google Terms of Service, the Google+ User Content and Conduct Policy, and the Google+ Pages Additional Terms of Service.)
Step 3: Be sure!

They obiviously mean it: You really need to be sure, because you are only getting one shot at this. Once you claim your custom URL you can't change or transfer it. And you can only have one URL to match your vanity.
Click on "Confirm choice". If you are hesitating, your last way out of this is the "Cancel". Then you can sleep over it and take another run at it in the morning..
Step 4: Congratulations! You have your vanity URL
On the lower part of your Goole+ "About" page, Google adds another section displaying your profile's custom URL and shows you for now a reminder to let everyone know about your new URL for your Google profile.

Make sure you leave your own Google+ Custom URL in the comment section and we'll circle you on our personal profiles as well as our Google+ Brand Page for +Relevanza.
Compare Google+ Custom URL with other social networks profile IDs:
The new Google+ profile URL: http://www.google.com/+BirgitPauli-Haack
Twitter Profile: http://www.twitter.com/bph
Facebook Profile: http://facebook.com/birgit.pauli-haack
LinkedIn Profile: http://www.linkedin.com/in/birgitpaulihaack
Twitter's profile ID is much shorter but that's mostly due to my 3-digit handle. LinkedIn provides a much longer profile URL.
Change to single post URL
What I really like about Google+ is that each post has its own URL. I sometimes use it to share Google+ posts on other social networks, especially when a discussion on the particular topic takes place.
At this moment, Google still provides the link with the large profile ID number, but that can be fixed quickly with a little copy/paste.
The provided link:
https://plus.google.com/101776844630674138344/posts/9ConzNqi4L8
becomes:
https://plus.google.com/+BirgitPauli-Haack/posts/9ConzNqi4L8
We have not yet received invitations for our brand page nor for our client's brand pages. As everyone else, we have to wait until Google decides to role out the custom URL feature to all brand pages. Google might select the next round of release from claimed business on Google Places and Google+ Local, which would make sense as those businesses already went through an extensive verification process. We'll update you as soon as we know more.
