Case in point, a few years I was on the home stretch of a 50 mile weekend bikeride, less than a mile from home. I was tired, dehydrated, and fuzzy-brained, but still alert enough to be paying close attention to the traffic around me. I was riding ~10-15mph in the far left lane of one-way 4-lane primary thoroughfare (35mph speed limit).
4-way intersection ahead, figured I'd cross the intersection and ride up onto the very wide sidewalk on the far side so I could get out of traffic, slow to a crawl, relax, and take it easy the rest of the way home. As I came up to the intersection I angled slightly toward the sidewalk ramp on the other side, then turned my head around to the right to watch the traffic coming up behind me to make sure no car was going to make a left turn at the intersection and either turn into me or cut me off.
All clear with cars, no left turners, but when I looked ahead again I saw I'd miss-judged my path and was going to miss the on-ramp by about 5ft and hit the nearly 1ft-high curb instead. Going too fast to correct course and hit the ramp, or to slam on the breaks and stop, only thing I could do was try jump the bike, or at least the front wheel, over the curb.
Too tired, botched the maneuver, not quite high enough, front wheel hit the curb about halfway up and went from around 10-12mph to 0 in a millisecond. Of course me and the rest of the bike kept going, me slingshotting over the handlebars (bike shoes clipped in so angular momentum at work here) and then piledriven head first into the concrete sidewalk. My head landed first with a loud crack, then my body landed on my head, then the bike on top of my body.
I might have been out for a few seconds, not sure, but the first thing I remember was thinking, "ok don't move a muscle, just lay perfectly till, you might have just broken your back and even the slightest movement could cause a tear in your spinal cord." Slowly, gradually, tentatively I started testing that individual fingers and toes worked, then hands and feet, then arms and legs, and finally body and neck.
Everything was ok, but I was still lying there not daring to try to get up. Next thought was, "ok, that loud crack was my head hitting the concrete, I probably have a concussion. gotta be careful, don't black out." Reached up to my head, started feeling around. Helmet broken in places, totalled as designed, but still attached, strapped on, conformed to my head.
Still lying on the sidewalk with bike on top of me, afraid of further injury, and feeling around my head checking for injury. No blood, no pain anywhere. Disbelief. Check again, thoroughly. Still no blood, no pain. More disbelief.
Can I really have just had a crash that bad, where my head alone absorbed the entire impact of my 200lb self and ~20lb bike, without any injury besides skinned hands and some bruises? Mindbogglingly, yes that was exactly what happened. That helmet absorbed the whole damn thing, and I walked away without either a concussion or any kind of neck or spinal injury, just some scrapes and bruises.
So yeah, just wear a helmet. Even if you have 10 accidents where a helmet is irrelevant and just 1 where it is, it will have been worth it.
If my anecdotal evidence doesn't impress you, then think of it terms of Expected Payoff E(X). Wearing a helmet costs nothing of consequence, yet the benefits could range from 0 (you're never in a crash your entire life and never need it, or you're in a crash where a helmet makes absolutely no difference) to everything (you're in a crash that would cause permanent brain damage or death without a helmet protection). You can see how the averages work out.
Humans are very poor by default at evaluating risks that have low odds of occurring, but very high consequences if they do occur. Bike crashes are exactly this kind of risk, so don't be a typical human here - don't blow these types of risks off, discount them, or ignore them. Be smart, mitigate or neutralize them.
You mentioned you obey the law so you probably don't do this. Even if the law does not prohibit riding on the sidewalk, you should never ride on the sidewalk.
Just wanted to take this time to remind everyone that bicycles do not belong on the sidewalk.
I'm a cyclist and I'm constantly looking out for them and I still sometimes accidentally walk into them.
One of the biggest problems with New York's new bike lanes is that a great many pedestrians treat them as sidewalks and cops don't give them tickets for it. It totally should be a jaywalking ticket.
That said, perhaps it's situational. My city has large amounts of arterial roads with heavy car traffic and sparse pedestrians. The speed limit on those roads is about 60 kph (about 40 mph), and I much prefer to stay off them as a cyclist. It would be much safer if I was allowed to cycle on the sidewalk in those areas, as there's little to no foot traffic. In the city itself, different matter. There's much more foot traffic, but the speed limit is lower, so perhaps it's best there to be on the road.
I am not saying go crazy and ride your bicycles on freeways. This is more about city streets. That being said, I truly believe the way to have fewer cars striking cyclists is to have more cyclists assert themselves on the road.
I'm also an advocate of taking up a full lane, but the laws in many places (like New York) actually require you to ride to the right/shoulder where feasible. Unfortunately that's not really clearly defined, so even though you have a legal right to a full lane, cops will still give you a ticket.
Having narrowly avoided being doored a few times, I won't ride anywhere within a foot's range of a completely open car door.
Sometimes it's the only sane option. While I can legally ride my bicycle on a four-lane road with a speed limit of 45MPH and no real shoulder, I'd have to be crazy to prefer that to the sidewalk that sees perhaps one pedestrian an hour.
The vehicle was a van, and turned right across me going straight (which normally I'd see clearly and avoid, but the way they hesitated implied they'd seen me so I rolled through. Alas.) I'd wager I was going around 20mph/30kph as I hit the side panel. Hips/shoulders first, then I felt my head whiplash into it. Without a helmet, I'd expect at least a mild concussion. With a helmet, I was able to roll away relatively unscathed (minor cuts/bruises) after we exchanged details.
Helmets can help at all/many speeds. They're not a perfect solution, but they're great sometimes. Don't discount them so bluntly :)
The helmets today are only tested with straight on impacts: they're tested by dropping them straight down on the crown of the helmet. But that doesn't happen in the real world. You're moving, meaning you're hit at an angle, meaning that there's a potential for a diffuse axonal injury[1]: the rotational component of the injury. MIPS[2] has helmets that help protect against that.
Is wearing a helmet safer than not wearing a helmet? Without a doubt.
Of course, if you weren't wearing the helmet, the truck mirror may or may not have touched you at all.
Had he not been wearing a helmet, that never would have happened.
I'm not saying "all". But I drive about 400 miles per week (long commute) and I can say that I've seen approximately 5 bicyclist last month which stopped at the stop sign. They really stand out of the crowd.
I've seen brains on the street from at least two low-speed bike accidents where the cyclist was wearing a helmet.
Also, it's not that all cyclists are disobeying the law. Those are just the ones you choose to see. Drivers and pedestrians alike just aren't looking.
I always obey traffic laws on my bike.
Edit: All these anecdotes are great and all, but it's not like we're going to be reading stories from all the people with brain injuries or who died. Cycling lanes and infrastructure do significantly more to make riding safer than helmets and helmet laws do.
http://www.bicycling.com/sites/default/files/uploads/BI-June...
http://greatergreaterwashington.org/post/19036/feds-will-sto...
http://bicyclesafe.com/helmets.html