Although I'm aware this doesn't answer the question, I think it's worth mentioning that laboratory research in the past 5 years has identified some of the reasons why antioxidants aren't as beneficial as hoped. Some of the clinical trials from NCI's list (from ghoppe's answer) were aborted after a iatrogenic rather than the hoped therapeutic effect of antioxidants was observed.
Why anti-oxidants had the opposite effect on the development of some cancers baffled people a long time... or more precisely until 2012, when a mice model of lung cancer was shown to be accelerated by anti-oxidants. And in 2015, anti-oxidants were found to promte metastasis of human melanoma (again in a mouse model); there was in fact a second study by a different group on the exact same topic in the same year. The findings were in line with the knwolege that the anti-metastatic drug methotrexate has some pro-oxidative effects.
So, the current picture seems to be that ROS is sometimes cancer-promoting (in the initial stages--if we stick to the conventional assumptions) but also prevents rapid tumor growth and spreading, whereas antioxidants have opposite effects, at least in some cancers. This is obviously an area of active research so more will probably be known in the upcoming years. It's also an area of high-stakes research because metastasis is the main cause of death in cancer patients.
A 2016 review ends with this advice
the multitude of studies discussed here suggest that patients diagnosed with (or at risk for) cancers should avoid unnecessary supplementation of antioxidants in order [to] ensure that enhanced antioxidant activity does not inadvertently facilitate tumor progression and metastasis.