I covered quite a lot of ground, but approaching 36 my knees are starting to give me troubles. Should I expect trekking poles to restore my previous stamina and walking durability? Or are they just fancy status symbols? :)
Here are some scientific papers, with my brief summaries.
Summary: They make hiking less damaging to your body, but they increase exertion.
I've heard rumors of studies that show that trekking poles reduce the force on your legs during a hike (by transferring it to your arms). The only reference to a study I could find was this one, but I wouldn't know how to verify that the study was well constructed.
This showed about a 25% reduction in the strain on the leg muscles of hikers (measured by an analysis of chemicals in the muscle tissue, and also the precived amount of muscle soreness from the hikers). The hikers trekked up and down a mountain trail in the UK, so it seems like a fairly realistic test. The study measured exertion of muscles, not knee injuries, but I would assume that reducing muscle strain would also reduce overuse injuries.
My totally antectodal experience is that 100% of the people I know who hiked long distances on the Appalachian Trail (either through-hikers, or guys that hiked a long portion) used trekking poles, and swore by them. They claim the poles reduce the strain on their knees, let them hike faster, and made them a little more stable when descending a trail. This included one 60-ish hiker, for whatever that's worth.
Also, I have no way to back this up, but I would speculate that anything that made you more stable on your feet would reduce the risk of, say, slipping and tearing your ACL while descending a loose trail.
No Silver Bullets
Of course, if you're having knee problems while hiking, you should think of trekking poles as part of your strategy to address that. Other things may include, looking at your shoes, maybe losing weight (if you're overweight), maybe adding in some lite strength training and stretching for your legs, reducing pack weight, and maybe even picking less aggressive hiking goals.
There are people who swear by poles, and there are people hiking into their eighties without poles or knee trouble. Part of it is genetics, part of it is being intelligent about hiking. Poles may help with the symptoms, but wouldn't it be better not to cause the problems in the first place? If you're having knee problems, look at your pack weight. If your dry pack weight is high (say, 20+ pounds) work on reducing the weight. If you're not wearing the right shoe for long distance hiking (trail running shoes with a little padding) that's going to be hard on your knees.
Buy or borrow a copy of Trail Life and read through it. I read it a couple years after getting back into hiking, and making the changes outlined in older versions of this book made hiking much more fun. The author Ray Jardine and his wife have hiked tens of thousands of miles through-hiking the AT, the PCT, and the CDT, among others.
Trekking poles are extra weight, and therefore will require extra exertion. Furthermore, they're extra weight on a moving part of your body, just like shoes are, which further increases the importance of keeping them light. Fix the other issues first, if any, so that you're not just treating the symptoms instead of the cause.
Trekking poles are a great helper if you have problems with your knees after a long march. A pain 'in knees' is usually a pain in the muscles around the knees, that are responsible for maintaining equilibrium. Those muscles are not very active if you walk on footpath, but in outdoors they are intensively used. With trekking poles, the equilibrium is maintained by providing additional fulcrum.
I can say with my experience, that using trekking poles have enabled me keeping pace with my comrades in mountains, which without poles were very hard for me, and additionally I had problems with knees.
But mention also, that using trekking poles will train your arm muscles, but will not train that muscles that are making problems by you. So you can become trekking-pole-dependent. After increasing my stamina with trekking poles, I've started to train walking without them, at begin in plains, than in mountains. Now I use them very seldom.
Also from my experience, trekking poles are decreasing fatigue of leg muscles, but increasing the fatigue of arm muscles, and overall you use more energy. I was able to increase distance walking without trekking poles. But if my knees are saying that they are too tired, I switch to trekking poles.