AAISP are the ones usually quoted as being the absolute best, but you will have to pay for it.
I was about to say this. We're with them and wouldn't dream of moving. Yes, they do cost a bit more and help is only available on the phone during office hours (although they often hang out and are contactable online on various platforms as they're all geeks - they have a huge dalek in their office!), but they have no call centres as such - just UK offices where a highly techy person who knows what they're talking about and actually wants to help you sort it will assist you based on whatever level of techy knowledge you do or don't have. Having said that, we joined them in 2011 and, other than setup assistance and a time where I made a very foolish error myself, I haven't needed to call them for the last 6-7 years. It just works. No broadband company can guarantee perfect uninterrupted service, but their logs usually show uptime of at least 99.9% of uptime each month.
They monitor your connection (NOT the content) every second and will contact you if they detect an enduring problem. Getting texts from them is quite a handy way of knowing if you've had a power cut when you're away from home.
They don't have any throttling, monitoring or censorship controls (you're very welcome to set up your own for your house). They use BT wholesale lines (I think you can also choose TalkTalk too) for backhaul, but they're very well-known in the industry for their knowledge and tenacity i.e. they will contact BT, tell them exactly what their engineers need to do to fix the problem, and hound them until they do.
I think it speaks volumes that they have a very clear 'How to leave us' link on their homepage. In my experience, companies who are so obliging in helping you to leave them, should you wish to, are the ones who know that very few people choose to.
They can provide a line for you if you're happy to make all/any landline calls through VoIP. If you want to keep PSTN/POTS (traditional landline calls) facility, then you'll need to rent the line from elsewhere (we're with Phone Co-Op, as recommended by a PP); also, they don't do any TV services, so you'll get a cheaper bundle from elsewhere, if the very lowest price is your priority rather than reliability.
I don't work for them, but am just an extremely happy customer who is highly unlikely to ever consider leaving them.
If you haven't already done so, check out ispreview.co.uk for a good overview of the market with plenty of helpful reviews. Obviously, they're subjective, but it's an interesting comparison to see the list of the 10 biggest ISPs (by customer base) alongside the list of the 10 best-rated ISPs. In the latter list, with the possible exception of Zen (which are usually well-rated) and Phone Co-op, 99% of people would never have heard of any of them. Also, be aware that big-brand ISPs (from supermarkets etc) are usually just one of the big boys with exactly the same service but with a household-name wrapper on it.