Things haven't really changed here, either - at least not when it comes to the standard 'shared student house' format.
However, there are a couple of other postgraduate accommodation options that may not have been available to you before.
Shared rentals
As with halls, you probably have some experience of renting 'third party' student accommodation during your undergraduate degree.
In principle, there's no reason why you can't go back to this option as a postgraduate. Landlords won't care what course your studying. If anything, they'll probably prefer letting to older, more experienced tenants.
The potential limitations of private rentals for postgraduate students aren't that different to those for university halls.
You'll still need to make sure your lease runs for 52 weeks (or have an alternative option for the summer). You'll also need to think more carefully about location and (though you will have more locations to choose from).
Finally, you'll need to find other students to live with. Unlike halls, you'll be able to do this for yourself. So, with care (and perhaps a bit of luck) living with undergraduate housemates doesn't have to mean putting up with undergraduate house parties.
Of course, this assumes you have the time to select suitable housemates - or the means to find housemates at all. This isn't always as easy to do when you're starting out at a new university.
Solo rentals
Living alone may not have been part of your undergraduate experience, but, depending on your background, it could be worth considering as a postgraduate.
You may find that a solo rental is more affordable now, particularly if you're returning to study after a period in work, or if you've managed to secure more generous funding such as a Research Council studentship.
If you can afford your own rental, this option has some obvious advantages: you'll have free reign over the format and location of your housing, plus the privacy (and peace) to get on with your Masters or PhD work without risk of interruption.
You might even be able to keep a cat, or other 'research assistant'.
Longer distance commuting
Living outside your university city isn't usually a desirable option for undergraduates. For one thing, the frequency of classes and assessments makes commuting difficult (and expensive). For another, most students are keen to take advantage of the independence and new opportunities that are part of 'moving away to university.'
Well, as a prospective postgraduate, you've probably done 'independence'. In fact, you may be quite used to living away from your parents. Chances are you'll also have less timetable now - particularly if you'll be studying for a PhD.
All of which means that living further away from your university may not be unrealistic, particularly if you only need to be 'on campus' once or twice a week (or less).
Of course, your mileage may vary (no pun intended). Some subjects do require postgraduates to be in regular contract with their institution and universities may actually expect full-time students to live within a certain distance.
Living as a resident tutor
Strictly speaking, this option belongs with the section on university accommodation, above. However, I've included it here because it's quite different to the standard version of 'living in halls' - and almost certainly wasn't available to you as an undergraduate.
Resident tutors are postgraduates or academics who live in undergraduate residences. It's their job to provide mentoring and support to students living away from home for the first time - and to ensure that good conduct is maintained within the accommodation.
Additional duties can include organising communal meetings and providing any necessary safety training or orientation.
These opportunities are sometimes offered to suitable postgraduates, as well as academics. In return for carrying out your responsibilities, your accommodation will normally be provided free of charge. Some roles may also pay a salary.
Needless to say, a 'free' room on campus is an attractive offer, but acting as a resident tutor will add to your workload. It can also limit your flexibility if you're expected to keep specific contact hours for the students under your charge.
Still, at least if your neighbours are noisy you'll have the authority to quiet them down. . .