Currently we're running a huge experiment in which virtually everybody is exposed to the electromagnetic fields of communication networks. This has been going on for more than a decade. If there is an effect, we should easily see it.
Note, I'm not saying there is an effect, but it's not quite so easy to say we should have found it if it exists. First, there is a delay between radiation exposure and cancer risk (up to ten years).
Second, if all population centers experience the same increase in radiation exposure, then it is very difficult to use radiation as an independent variable to measure its impact on cancer.
For instance, it took decades before anyone was able to show that nuclear testing caused an increase in deaths, and that was only possible because the distribution of radioactive isotopes was not uniformly spread across the US -- some states (especially in the midwest) received far more radiation than others (e.g. Western California or Louisiana).
In contrast, if electromagnetic radiation from cell towers is relatively uniformly spread for the same population density (e.g. New York is comparable to Los Angeles, and rural Washington is comparable to rural Missouri), then it will be rather difficult to separate cancer risk from electromagnetic radiation from other risks.
Edit: To make matters worse, there is even a profesional disease among people with RF exposure: eye cataract, due to thermal heating of a poorly irrigated and termoregulated corneal tissue. These are guys working in close proximity to megawatt transmitters. It is not correlated to cancers.
Clearly, but anybody claiming the spooky action effect needs to accept the burden of proof and produce positive evidence. When that necessitates new ways of physical interaction unknown to science, it's a colossal burden, likely to have far reaching implications and revolutionize multiple fields; it's highly unlikely that such positive evidence was ignored for the decades the effect was claimed.
However, the field strength generated by everyday devices are regulated to levels that are thought not to generate a relevant heating effect in typical use.
Additionally, looking at your phone means it is an appreciable distance away from the brain, unlike using a phone as a phone.