The biggest fallacy for me is the 'kids must wear helmets' fallacy.
This is the fallacy whereby schools periodically have 'ride to school days' or similar and they dictate 'helmets must be worn'.
Which is all very well but kind of ignores the fact that
(a) the kids have no road sense
(b) the bikes have not been ridden in months and are basically death traps
(c) the bikes came from Toys R Us and are not fit for the purpose in the first place.
There really is nothing worse than the attitude that bike safety is solely, or primarily, about wearing a helmet. They had to edit helmets into a Peppa Pig episode for reruns because she didn't have a helmet on.
There are so many dangerous things that cyclists, both adult and children do, that focusing on helmet wearing is absurd. It's like saying to someone doing 100mph in a residential area that 'you should wear your seatbelt'.
Helmet wearing has a negative to minimal impact on cycle safety. If people want to wear one, fine. If they don't, likewise. But please, rather than fretting that your child is riding to school with no helmet, why not observe them as they cycle from the pavement into the road, down a one way street the wrong way, etc. And take action.
You should buy, and read, this:
www.amazon.co.uk/Cyclecraft-complete-enjoyable-cycling-children/dp/0117037400/
before you even consider this:
www.argos.co.uk/static/Product/partNumber/3333516/Trail/searchtext%3ECYCLE+HELMET.htm
Of course this won't happen because it's more convenient to blame cyclists for being injured based on wearing/not wearing a plastic hat.
And please ensure that your tyres are safe and properly inflated (this is very difficult to achieve without a proper track pump with gauge costing £15 or so), that brakes are adjusted and effective, forks around the right way, that you have effective luggage (backpack, or better, basket/rack and panniers), rather than dangling things off the handlebars.
All these things have an unambiguous safety benefit and are very commonly not done, yet people still blather on about helmets?