Randomized clinical trials with placebos are the gold standard of science. They are used to generate generally effective results for pharmaceuticals and many herbal supplements, although the way a trial is designed can have a great impact on whether the results are truly reflective of the reality. Some of the trials for herbs have come under fire in particular for using insufficient amounts of herb, or the wrong type of herb. And pharmaceutical companies have sometimes used the variability in these trials to obtain the results they want.
However, a well designed trial is generally a good indicator for these products, which are not otherwise things we consume on a regular basis. And these things generally have specific parts of the body they exert an effect on, which can be more easily measured in a randomized trial with a placebo.
Unfortunately, this is not necessarily the case with vitamins and other nutritional supplements. An article in Nutraingredients summarized very well the limitations of these clinical studies for establishing whether vitamins and nutritional supplements prevent chronic disease.
The author, Stephen Daniells, makes the point that chronic disease has a long latency period. Yet the trials only follow people from 2 to 7 or so years. One example cited was a study looking at the possible impact of vitamin D and calcium on colorectal cancer. Colorectal cancer takes 10 to 20 years to show up in healthy subjects, which the women who took part in the study were. Yet the study only followed them for 7 years before coming to the conclusion that these supplements had no effect. This is not good science.
Another point Stephen makes is that vitamins, minerals, and dietary supplements like antioxidants work on many different parts of the body, and they work synergistically with each other. This means that some of them, when used in combination, actually produce an effect greater than the individual sum of their parts. Combine this with the need for very long term trials, and you have something of a difficulty with randomized clinical trials for vitamins and supplements.