Enemalta supplies electricity, WSC water. They are billed through ARMS, and there are two tariffs, residential and domestic. The domestic tariff is 30-50% higher, so its clearly better to be on the residential tariff. You can tell what tariff a residence is on by looking at the latest bill - if the number of residents is 0 its domestic, if 1 or more residential. The more residents shown the better.
Domestic tariff is for residences (and things like garages with a meter) where no resident of Malta lives as their primary dwelling. That means this rate is for 2nd homes, or for properties rented to, or owned by tourists. This is a contentious issue, and some foreigners believe its discriminatory against foreigners, whereas the government says its based on residency not citizenship, so is permitted under EU law. There is a court case pending to clarify if this discrimination is legal or not.
If you buy a property in Malta and its your primary home, and you get the new eResidency document (or have a still valid old residency certificate), you just need to fill in the ARMS form and all should be fine
If you rent property, then its somewhat harder. There are two options -
1. have the bills in your name, or
2. have them in your landlord's name, but for your residence to be accepted by ARMS.
Having the bills in your name means having to pay a deposit of around € 500 (whether you are Maltese or foreign), so most go for the 2nd option. Either option requires your landlords permission, so its something to discuss before signing a rental agreement. Some landlords wont agree as it puts them on the taxman's radar. In either case, you need to prove you are a resident of Malta, and the simplest way of that is by showing ARMS your new eResidency document (or have a still valid old residency certificate). ARMS have accepted other forms of proof, but its at their discretion.