During the summer, I like to go for rides on trails without a shirt. Is this safe?
Also, related: is it rude to other people on the trail not to wear a shirt?
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Safe is a relative term in this instance. A shirt might protect you in a fall, but only just. You'll know immediately whether a shirt can protect your skin from sticks, branches, and prickers dangling into the trail. If you were going fast enough flying insects may even make an impact, though I can't imagine it'd be terribly painful.
Shirts do offer some protection from two things: sunburns (and subsequent skin cancer) and bites from insects (potentially carrying disease.) For lightly complexioned people, skin cancer is a huge danger and can develop even without having been sunburned. Insect-borne diseases are also a big deal in most places. Having been treated for Lyme disease, I can attest to the importance of wearing insect repellent and, yes, shirts.
Additionally, that thin layer of material actually keeps you cooler by blocking the sun from reaching your skin - this is the same reason why desert dwelling peoples wear lightweight material all over their bodies. Having a shirt on also keeps you from sweating all over your bike and other gear.
In terms of etiquette: if you're out in the woods and there aren't many people around, go for it. If you're bumping elbows with people on a multi-use trail, maybe consider some modesty.
Bit of a personal opinion on whether it's rude. Personally I'm not bothered at all. I'm more offended by dry creaky chains. I think legally (in the UK, at least), it legal for anyone to show their bare chest in public, regardless of gender.
A sleeveless cycling jersey offers no protection to the arms and might save a scratch or two in a light tumble. It might save some gravel rash on the body. It does offer protection against sunburn - but sunscreen the upper arms!
But for a shorter ride or say doing intervals on a warm day I find riding shirtless feels great so take the risk
A typical shirt or bike jersey has no security benefits. Depending on the terrain some special protectors might make sense when thining about security. One might even argue that a shirt might reduce security as it might be caucght by a tree or something Whether iit is "rude" depends on your local environment. In some areas it's absolutely accetable, in others not.