Is it possible to scientifically prove or disprove the existence of god(s)?
In general, it is extremely difficult to prove a negative. In fact, if one adheres to the scientific method, it is not even necessary. The burden of proof is always on the person who makes a positive assertion. He must provide supporting evidence in the form of observation and/or logical reasoning before anyone is obliged to treat his assertion with respect. Once he does that, then in accordance with the scientific method we can peer-review his evidence or reasoning. If we succeed in falsifying it, with a community of peers looking on to make sure we did it honestly, competently and scientifically, then he's back where he started, with one of the millions of unsupported hypotheses that we ignore every day.
In other words, we can't disprove the existence of gods, but we don't have to since it's an assertion with no respectable evidence. In just the same way, we don't have to prove that dogs can't fly, because there is no evidence that they can.
Or, if not prove, at least provide some evidence one way or the other.
To prove the existence of gods, one must start with the presentation of evidence. I will digress here and explain the Rule of Laplace, one of the cornerstones of the scientific method: Extraordinary assertions must be supported by extraordinary evidence before anyone is obliged to treat them with respect. (Notice that we are not
obliged to mock them, but we are
free to do so, which is why you see so much of that mocking on SciForums.)
Now the scientific method, indeed all of science, is derived from the premise that the natural universe is a closed system whose behavior can be predicted by reasoning logically from empirical evidence of its present and past behavior. To assert that a supernatural universe exists, full of invisible, irrational creatures and other forces who have the power to capriciously interfere with the behavior of the natural universe, is to assert that the scientific method is false, and is therefore the granddaddy of all extraordinary assertions. The reason is that the scientific method is
recursive; it has been tested by its own methods for half a millennium, and it has never come close to being falsified. So you can easily see that for someone to claim that it is, indeed, incorrect, he is going to have to come up with some pretty extraordinary evidence. The typical "evidence," such as one of the tens of billions of tortillas that are sold every year having on it a splotch that is said to be the likeness of a biblical figure,
of whom no portraits exist against which to compare it for accuracy, is clearly not extraordinary evidence.
If you think it is possible, how would you go about it?
The scientific study of the natural universe is not complete. Things occasionally happen that defy our attempts to explain them scientifically. Abiogenesis is an important example which is in the news. We have some very good ideas about how the first living matter developed naturally from non-living matter, but there are still parts of the process that we don't understand.
The best way to falsify science would probably be to start with one of these high-profile enigmas and look for evidence of supernatural causes. Note that simply saying that we haven't found a natural mechanism is not the same as proving that there is a supernatural mechanism. Both sides in the controversy still have to do their homework. The first side that finds evidence for their assertion shifts the burden of proof to the other side: they must falsify the evidence.