Since this question is about telepathy, I'll focus on just that and also use the definition that it is,
... the transmission of information from one person to another without
using any of our known sensory channels or physical interaction.
First, the major scientific criticisms generally boil down issues with the following points:
To address the first point, experimental design is a major concern when any tests for telepathy are examined since they must be controlled in such a way that the observer-expectancy effect is controlled for. Any tests also need to be conducted in such a way that subconscious cues to the subject are controlled for to prevent the "Clever Hans effect" or cold reading.
Lack of a model as to how telepathy would work is actually a major problem for any proper study for telepathy since it exposes the experimenter to criticism if you try to explain null results as being due to the subject being in the wrong state of mind or out growing their abilities (Horn, 2009). This exposes testing to major issues with reproducibility and lack of reproducibility previously positive results can easily be explained as problems with the experimenter or attempts to defraud the experiment as opposed a potential explanation for why telepathy might not present on a consistent basis in laboratory conditions.
Ultimately, Joseph Banks Rhine is likely one of the few scientists to have studied telepathy in controlled environments that might have presented some evidence of an effect (Rhine, 1964); however, even his work has been subject to major criticism and generally his work is dismissed by most skeptics. The extent of his research does raise an interesting point though that relates back to the lack of a working model: namely, is it even possible to dismiss unreproducible experiments on the basis of a single experiment or are multiples needed before results can safely be dismissed? This does raise the issue of publication basis though since a researcher can be accused to not publishing null results while someone that is seeking to reproduce the results could be accused of the same by not publishing positive results.
In conclusion the best we can really say is that there is weak evidence that there might be telepathy; however, there is a lack of explanation of how it could occur or that Littlewood's law1 isn't at play during experiments and lack of reproducibility ensures that the current scientific consensus is that telepathy likely does not exist. However, this is not to say that "synthetic telepathy" is not possible since brain-computer interfaces are in development that mimic what people generally think of when they define telepathy.
That said, one of the statistical meta studies linked in the accepted "No" answer had the following to say:
The recent focus on meta-studies in parapsychology has revealed that
there are small but constantly non-zero effects around studies,
experimenters and laboratories. The sizes of the effects in
forced-choice studies appear to be comparable to those reported in
some medical studies that had been herald as breakthroughs.
Which can be taken to imply many different things; however, their conclusion also serves as a good one for this answer as well,
If ESP does not exist, there is little to be lost by erring in the
direction of further research, which may in fact uncover other
anomalies. If ESP does exist, there is much to be lost by not doing
process-oriented research and much to be gained by discovering how to
enhance and apply these abilities to important world problems.
Horn, Stacy (2009). Unbelievable: Investigations into Ghosts, Poltergeists, Telepathy, and Other Unseen Phenomena, from the Duke Parapsychology Laboratory. New York, NY: HaperCollins Publishers.
Rhine, J.B. (1964). Extra Sensory Perception. Boston, MA: Branden Publishing Company.
Rhine, J.B., Pratt, J.G., Stuart, C.E., Smith, B.M., Greenwood, J.A. (2011). Extra Sensory Perception After Sixty Years: A Critical Appraisal Of The Research In Extra Sensory Perception. New York, NY: Henry Holt.
- Littlewood's law, or adage, states that an individual can expect to experience "miracles" at the rate of about one per month.