Hi -
I completed the process a few months ago and, for an EU passport holder, it was pretty painless and required two steps. Note my wife and I obtained residency in Madeira and your experience will probably be slightly different than mine.
1) I went to the city hall and provided some information.... one key document was a lease or proof that I was living in the parish (you have to apply in the parish you are residing). That, along with passport and other documents. We over prepared and brought along proof of health insurance, FBI background checks, etc. After successfully completing this step and paying 15 Euros, you will receive a document basically indicating that you have proven that you are living in the Parish.
2) Then, we went to another office, across the square, and provided much of the same information plus the official document that confirmed our residency. After working with a nice staff member who helped us with the form, we waited 2 weeks for an email indicating approval of our residency permit. We returned to city hall, We paid another 15 Euros (i believe) and walked away, delighted, to have our residency papers in hand.
I'm not sure what others received, but we received a paper document, not a residency "card" that would fit our wallets.
Hope that helps....