For basements and other small size landlords, pets can be a big problem. But in large buildings, they can't prevent you having a pet like a cat or dog or much of anything else, provided the pet does not do any damage to common areas, or the unit, and does not bother any other tenants.
Pets not being allowed has been a standard clause in most apartment rental contracts forever, but there was a case that went, and I can't recall just now whether it went to the Supreme court or Human Rights, over an elderly lady who had a cat in a no pet building. She'd had the cat for years, nobody knew, 'til one day the super caught sight of it. This was back in the early 90s IIRC.
The woman was given a choice, get rid of the cat or be evicted. She wouldn't get rid of her cat so they evicted her, even though she hadn't anywhere else to go, and it meant she'd end up having to pay a lot more for rent in a new place.
The case generated so much publicity, all of it bad for landlords, because it seemed so heartless. Elderly lady, all alone, just her cat, and if nobody even knew it was there for years, the cat was certainly not bothering anybody, nor doing any harm to property. I'd have to try and look it up to be sure how it was resolved exactly, but she ended up with a place to live and her cat, and the no pets clause ended up becoming one that was unenforceable in Ontario. Provided the pets cause no damage and don't bother other tenants you can have dogs or cats, but municipal bylaws may limit the number of animals you can have. Some reptiles aren't allowed, some exotics aren't allowed, and you can find out from City Hall exactly what those are in any given city or township.
Condo's are an entirely different story because they have both bylaws and rules, which can differ between every separate condo corporation. If the rules say no pets, it takes a majority of registered owners to vote to change a rule like this. So you do have to ask up front, because it can cause you to be evicted for cause if the condo becomes aware of an illegal pet. The owner of the condo can find themselves in the position of having their unit seized and sold in some cases. I'm not any kind of expert on condo law, but I have a friend who served on a few condo boards and helped her write their rules for one of them, including the ones about pets. Their old rules didn't allow any dogs, but it was changed after months of lobbying owners to ask for a vote to change this rule. Since you're basically renting a house from its owner, they can pretty much make any restrictions they want to make. It may not be strictly legal, in terms of the Landlord and Tenant act but they can simply refuse to rent to you and fighting that kind of thing is far too much trouble to be worth the effort. Same goes for basement apartments, pretty much.
It used to be you would get a big hassle from landlords about water beds. Many would not allow them when they first began to be popular. In part that was because of worries over their weight, in part the possibility they might leak or cause a flood. It's not much of an issue now, unless maybe you're in a really old walk up flat with weak floors, perhaps. But you're often renting flats like these over storefronts, and the owners may live in one of them too, so some of them can be a bit problematic despite the legalities.
Big highrises can vary a fair bit, depending who owns them or who manages them. But having a very large tank is probably not the greatest idea in a rental situation. You have to have whatever liability and/or tenant property insurance you can get so you are covered in case of leakage or flooding, which might affect a number of units other than yours. Be very careful you maintain the filters, etc. properly, and are careful with buckets or Python type devices, so you can't be accused of negligence. It's also often the case with most types of home insurance, if you are away for any length of time, you have to have someone coming in regularly to make sure all is well for the insurance to remain in force in case of a problem. Nothing to do with whether the fish get fed, it's all about somebody checking that nothing is leaking or flooding, be it a fish tank or the water pipes.
Read the Landlord and Tenant act, find out what your legal rights are as a tenant - something most tenants never do but should. Find out what insurance is available so you'll be prepared with answers if/when the subject comes up. If you're prepared with the right knowledge and information it can be very helpful. Be honest with a potential landlord, as trying to sneak something by them can often backfire on you.