I came across a Software Advice article based on an interview with Ryan Singer, Product Manager at 37Signals. I used their Basecamp tool on a project not to long ago and found it to be a pretty good tool.
In the article, Singer talks about how they organize work on a project. Instead of assigning all the tasks by roles, they organize the work around what Singer refers to as projects...but I would call them features. Using their example, one of the projects (features) of a conference registration application would be a receipt page. They prioritize the work by each of the features or what they refer to as areas of concern.
To me is sounds like they are taking more of a Kanban approach, prioritizing the features and working on them one at a time rather than planning work around iterations. I also didn't see an reference to User Stories.
I took my PMI-ACP certification exam earlier this year, and in my studies came across a discussion of process tailoring in Mike Griffith's exam prep book and the concept of Shu-Ha-Ri, which I blogged about a while ago myself. It seems like the folks at 37Signals have done some tailoring that has worked for them, but is their approach right for everyone? Probably not.
In the spirit of Shu-Ha-Ri, you should start by following the rules. For most people, this probably means Scrum. From here, you need to figure out which rules to break for your organization. Researching what other organizations like 37Signals has done is good for ideas, but ultimately you need to figure out what works in your organization. As Singer points out, there are many creative ways to use Basecamp, just like there are many creative ways to tailor your agile processes to meet your organization's needs.