(Ex-LL)
Unless an LL can't let to anyone else, most will choose a non-smoking tenant without pets or housing benefit
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the lease at my last place specified no pets and no smoking indoors, even in communal areas like landings & staircases
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my LL insurance specified no pets and noone on housing benefit
I would have let to tenants with a registered assistance animal - exempt on the lease - and I told my agents this.
Like it or not, if you want to keep your pets, then you will probably have to offer a significantly higher rent.
Some LLs (like me) would risk it for a good tenant who'd been there 3+ years and wanted to buy an animal,
but your problem is you would be starting from scratch, as an unknown quantity.
Your promise to pay for any damage is not enforceable - especially if it is thousands that you may not have
and as pp said, offering an extra deposit doesn't help, as an LL usually can't claim full damages back.
When a tenant accidentally burned some flooring, I only received a small amount of what it cost me to put righjt,
because for that standard of property, I couldn't have an odd non-matching square
Also a significant % was deducted for the flooring being 3 years old - it was the high quality kind that normally wouldn't have needed replacing for years.
So, YABVU, if the law were changed to add additional risk of high expenses, particularly pets, many LLs would not let property.
Thousands of pounds worth of damage, plus months of lost rent during repairs, plus legal costs, could mean actually losing money that might take years of letting to regain.
btw, to other pp, smoke stink can linger for years and can require replastering and repainting, replacing fitted units, plus replacing furniture if let with that