Madeira has very little crime and the worst that is likely to happen to you is a sunburn or a hangover. Violence against tourists is very, very unusual. Common sense at all times should spare you any grief.
Pickpockets and bag snatchers are not a problem, however be aware of people that brush up close to you or jostle you and don't leave bags and camera equipment unattended for even a moment.
Avoid dingy, empty alleys and back streets in the 'Old Town' or anywhere else for that matter that doesn't feel right at night.
If you are the victim of theft, you must report it to the police and get a report copy, if you want to make an insurance claim. If your passport is stolen or gets lost, you must contact the consulate to get a replacement issued.
There are not many nuisances but noise can be a problem. It seems that Madeira is unaware of the term 'noise pollution' or of its effects on our health. Whilst the majority of Europe is investing in noise reduction, many of the local inhabitants actually spend money to make their motorbikes and cars sound louder. If it does get too noisy for you... just leave the city and head for the more tranquil areas.