What model should I use to find a common pattern for a specific user group based on the other dimensions?
The most common approach is to create business rules handmade, based on the univariate and multivariate analysis of the variable.

Basically, do some frequency count, see if you could isolate some subset of your data just looking at one or two variables.

Then when you have your labels, create a linear or so model with this new variable as output. For exemple, a linear discriminant analysis. The analysis will supply you with new insights on your group.

If you want to rely on an algorithm, two solutions:

As you don't seems to have a lot of variables, a non-supervised segmentation could do the job. For exemple, a k-Nearest Neighbor or a decision tree are basic and good aproachs.

With a few more variables, I like is to do a principal component analysis then a non-supervised classification to define your group on the result of the PCA.
Note that a PCA + handmade rules based on the analysis of your PCA result may be enough.

Each time, in the end, a discriminant analysis and a profile of your groups to asses the quality of your results.