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Cyan comments on The Statistician's Fallacy - Less Wrong
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Microwaves are almost certainly safe, but just FYI, the point about there being 'no plausible mechanism' is wrong, and a common misconception. Photons don't need to have enough energy to directly cause DNA breakage, in order to be dangerous. Microwaves seem have effects on proteins beyond that caused by thermal excitation, which means they could plausibly be carcinogenic, e.g. if they interfere with DNA repair enzymes. There's some evidence that pumping enough microwaves at cells in culture can turn them cancerous.
The epidemiological evidence though, is that they don't cause cancer.
http://www.ncbi.nlm.nih.gov/pubmed/11088227
Links?
All the stuff I've found makes the epidemiological evidence sound inconclusive, but the arguments from physics / biology seem pretty solid. But I've also heard people suggest that descriptions of the epidemeological evidence sound inconclusive because what they really mean is "if there's an effect, it's too small to detect," which scientists are afraid to say because that would also be misinterpreted. I'd really like to get clearer on this.
Querying my brain for specific sources turned up NULL, so I spent a couple of minutes on pubmed. It seems my statement was too confident.
There was a large metastudy which found some effect in a high quality subset of studies: http://www.ncbi.nlm.nih.gov/pubmed/19826127 And a commentary on it which says their definition of 'high quality' is bullshit, amongst other things: http://jco.ascopubs.org/content/28/7/e121.long
I find the criticisms in the commentary convincing, but I'd still rate their being a reasonable chance of an actual association existing, and therefore a non-negligible risk of an actual causal relationship as opposed to just some confound. I invite anyone who likes this sort of thing to give it a little more time.
The more obvious plausible mechanism for cell phones causing cancer, is that people with a certain lifestyle are more likely to buy and use a cell phone, or that owning a cell phone increases stress or somehow contributes to a different lifestyle, or some other mechanism that doesn't involve dim sources of low energy photons.
BPA causes cancer. Microwaves + plastics DO cause cancer. I'm just gonna put baking paper to line my microwavable containers before stacking food on top for easy cleaning and any currently unknown leaking hazards. Plastic safety seems like a protoscience at the moment.
The OP was talking about phones, not ovens.