I found that barbers were charging me the 'senior citizen' price, without asking me whether I qualified for it, not long after I was 50!
A few things are fixed but many are entirely discretionary. (e.g. Hairdressers can give you the lower price when you're 50, insist that you wait until you're 70 or simply not offer any discount at all).
For what it's worth though . . .
You qualify for free bus travel at the age when you first get your state pension (or, for a man, when a woman born on the same day as him first gets her state pension) in England. Elsewhere in the UK you qualify at 60. (Passes are only valid in the relevant country. You can't, for example, use a Welsh bus pass in England or vice versa).
You can get a Senior Coachcard (giving you a third off National Express coach services) for £12 (+£2 p&p) when you're 60.
You can get a Senior Railcard (giving you a third off rail travel) at 60. It costs £30 per year (online or from rail stations) or £70 for three years (online only).
Organisations such as English Heritage offer discounted membership fees from age 60. (With the National Trust though you also need to have paid the full rate for 5 out of the previous 10 years in order to qualify).
Museums vary. Some give a discount for people over 60; others insist that you're at least 65. However I've never known any to actually ask for proof of age anyway.