They have been tested and found to have no significant real effect, though they may have a placebo effect as people are told that they have positive effects.
In fact, many supplements are misunderstood. The book
Bad Science really shocked me. Basically, Ben Goldacre is a doctor in the UK who runs a website devoted to
bad science all about dubious medical science.
I can't remember all the details, but some things I remember from the book were-
Taking too much vitamin C can increase cancer death rates. When a study was conducted on lung cancer suffers taking extra vitamin C they had to end early because the death rate was too high in those with cancer (if I recall it increased about 40% over the control group!). It was immoral to continue.
The evidence that Omega-3 improves concentration and intelligence is quite dubious. A big factor in the recent craze was from a UK school trying to improve exam results. Over a year they made many changes, just one was getting people to take Omega-3. There was no control group, and results improved. Read more
here.
The book also mentions things like homeopathy, MRSI and the MMR vaccine.
It pains me as a science graduate to see people needlessly paying money for all these tablets. If your doctor diagnoses you with a deficiency, then of course take them. However, in general just eat a good diet with lots of fruit and veg, and bad things in moderation.
There are so many myths about this, people I know start popping handfuls of vitamin C when they get a cold. They don't realise that taking too much might be harmful, and once you get a cold upping your intake then is really too late. Provided you aren't deficient in vitamin C (you won't be if you eat fruit) just buy some tissues and suffer out the cold.