Do shopping centres or other similar locations in Dubai have free WiFi? I'll be traveling there with friends and want to be able to report in that we're alive after the hike.
Big malls have free WiFi for sure. The Dubai Mall offers it free of registration, while Mall of the Emirates require a simple registration - the password will be sent to your phone with a SMS.
The hotel where I was staying had free WiFi in the hall.
If you really need internet I suggest to get a SIM card from local company Du (Emirates Integrated Telecommunications Company) or Etisalat for the duration of your stay.
In addition to the answer by Alessandro, I like to add that (at least in spring 2014) the registration in the Mall of the Emirates used to require a LOCAL telephone number. This can be quite a problem for tourists.
If you are willing to pay for wifi, there are multiple options in Dubai, using credit card or skpe.
Wifi can be found fairly widely, the big problem though is that almost all of them need a UAE mobile number to receive your login details. Non-UAE numbers aren't accepted. Login pages often look like this, with the UAE restriction:
Assuming you do have a UAE number (/get one), then you can find wifi at most of the main metro stations, tram stops, various public spaces, museums etc. Most of the malls too. The Mall Of The Emirates (the one with the ski slope in it) is UAE-only, the Dubai Mall (Burj Khalifa attached) does unusually let you register with foreign numbers
Many western-style cafes have wifi, but you need to ask for a password, normally given as a one time ticket or scratch card. They often seem to have time or data restrictions, which mean you'd struggle to be there for hours.
Wifi that you can access without a password and without registration is very very rare in Dubai
Given that you'd have to get a UAE sim anyway to get online with most wifi, you might be better off getting a UAE sim + loading some cheap-ish data onto it and using that!