As a regular bike rider and helmet-wearer, I was surprised to hear serious substantial doubts about their usefulness (hat tip Brandon Reinhart). I’m pretty sure I’d want a helmet as a soldier or motorcylclist, but that doesn’t mean any helmet helps against any injury. The bike helmet wikipedia page is dominated by helmet skeptics, and there are several well-argued skeptic pages out there, such as this page full of studies.  Consider, for example, the effects of a 1992 helmet law in Western Australia:

Cyclist head injuries didn’t obviously fall a lot, and non-head injuries rose a lot.  There are lots of conflicting studies (e.g,. pro and con), but the best skeptics seem at least as competent as the best pro-helmet researchers.  Selected quotes from wikipedia:

Ordinary cycling is not demonstrably more dangerous than walking or driving, yet no country promotes helmets for either of these modes. … Six times as many pedestrians as cyclists are killed by motor traffic, yet travel surveys show annual mileage walked is only five times that cycled; a mile of walking must be more “dangerous” than a mile of cycling…” The proportion of cyclist injuries which are head injuries is essentially the same as the proportion for pedestrians at 30.0 % vs. 30.1 %. …

Robinson’s review of cyclists and control groups in jurisdictions where helmet use increased by 40% or more following compulsion concluded that “enforced helmet laws discourage cycling but produce no obvious response in percentage of head injuries”. This study has been the subject of vigorous debate. …  The largest [study], covering eight million cyclist injuries over 15 years, showed no effect on serious injuries and a small but significant increase in risk of fatality. … The head injury rate in the US rose in this study by 40 % as helmet use rose from 18% to 50%. …

When mandatory bicycle helmet laws were enacted in Australia, slightly more than one third of bare-headed cyclists ceased to ride their bicycles frequently.[58] In the UK between 1994 and 1996, in areas where cyclist counts dropped, wearing rates increased and where the number of cyclists increased, helmet wearing rates fell. …

A reduction in cycling may lead to an increased risk for the cyclists remaining on the road. … According to one source, the probability of an individual cyclist being struck by a motorist declines with the 0.6 power of the number of cyclists on the road. …  One researcher randomized his helmet use over a year of commuting to work and found that he rode slightly faster with a helmet. …  One small study from England found that vehicles passed a helmeted cyclist with measurably less clearance (8.5 cm [out of] … 1.3 metres) …

Cyclists’ representative groups complain that focus on helmets diverts attention from other issues which are much more important for improving bicycle safety, such as road danger reduction, training, roadcraft, and bicycle maintenance. Of 28 publicly funded cycle safety interventions listed in a report in 2002, 24 were helmet promotions. For context, one evaluation of the relative merits of different cycle safety interventions estimated that 27% of cyclist casualties could be prevented by various measures, of which just 1% could be achieved through a combination of bicycle engineering and helmet use. …

The following countries have mandatory helmet laws, … Australia, Canada, Czech Republic, Finland, Iceland, Israel, Slovakia, Sweden, USA, and New Zealand. … In the U.S. 37 states have mandatory helmet laws, and nearly 9 in 10 adults support helmet laws for children. Israel’s helmet law was never enforced or obeyed, and the adult element has been revoked; Mexico City has repealed its helmet law. … The countries with the best cycle safety records (Denmark and the Netherlands) have among the lowest levels of helmet use. …

Boston had far higher rates of helmet-wearing (32% of cyclists, versus 2.4% in Paris and 0.1% in Amsterdam), Amsterdam had far more cyclists (242 passing bicycles per hour, versus 74 in Paris and 55 in Boston). Cycle helmet wearing rates in the Netherlands and Denmark are very low. … Despite the lack of helmets, cycling in the Netherlands is safer than in any other country.

I plan to be much more casual now about whether I wear a helmet, especially since most of my biking is on mild trails away from cars.

Added 1p: Tim Harford considered such doubts and concluded bike helmets aren’t about safety.

I’ll keep wearing a helmet. I am a) Risk averse and more importantly b) Signalling to my wife that I pay attention to her opinion.

Added 8Aug2020: This review finds that helmets help.