After thinking about some of the points raised in the comments, I would like to expand on my answer, but also defend its form against the criticism that it is so vague as to be unhelpful. [In case you are wondering what the original answer was, it is roughly the sections 'Looking for mistakes' and 'Trusting your feelings'.]
This was mentioned by @EnergyNumbers. The answer is very popular; however I don't think that it is particularly helpful.
Benford's law is only one of many statistical techniques that can be and have been used to detect fraud or bias. It has become widely known, probably partly because it is simple to apply, but also simple to justify in a 'hand-waving' way.
However, its validity is much more limited than @EnergyNumbers (who calls it the unreliable way) implies. As originally formulated, Benford's law said that if you take a large range of numbers which have different sources, contexts, meanings or units, the logarithmic distribution emerges. This is a very interesting statement, but has little utility in detecting fraud. The statement that Benford's law, whether applied to first or second digits, should apply to a particular set of observations of a single variable, is an extremely strong statement. There are many, many natural examples of well-formed non-fraudulent datasets to which Benford's law does not apply. Several other digit distributions could reasonably arise in bona fide data. You may or may not be able to justify the assertion for your own data, however what you should not do is blindly apply Benford's law to various sets of numbers, and start forming opinions about their reliability.
It is a serious statistical technique and requires non-trivial statistical understanding to apply. The same thing applies to checking for normality. Unless you have a good understanding of how normal distributions arise, you will not be able to form a theory as to why some distribution should be normal. If this is the case, then any test for departure from normality will be useless.
A paper that really examines this for Benford's law is The Irrelevance of Benford's Law for Detecting Fraud in Elections. [Hat-tip to @Flounderer who linked this in his comment.]
Why this answer doesn't go into any statistical detail
The original answer I gave, below, tries to err on the side of not handing 'formulas' over to people who possibly don't understand their use. I tried, perhaps not very successfully, to suggest starting places for thinking about how and maybe why people either fake results or unconsciously introduce bias.
This kind of forensics is in some ways very similar to other stats, but has some very important differences. If you are looking for a signal in some noise, you might form two hypotheses, both of which imply that data is random, but with say different means or distributions. If you are looking for cheating, you have to remember that fraudulent data is not in any sense random. Spotting it involves teasing apart (possibly) three elements: the real numbers, the deliberate adjustment, and any pseudo-random perturbation that might have been made to mask the adjustment.
I believe that in order to properly apply some forensic test to a set of data, you need to first develop a proper theory of why the test might be meaningful. This entails hypothesing about exactly how the data might have been manipulated. For example, Benford's law was successfully used to investigate whether China's GDP growth in % was being rounded up if it had a high second digit: http://ftalphaville.ft.com/2013/01/14/1333552/chinas-non-conforming-gdp-growth/ (registration required).
Taking a whole battery of tests and applying them to some data might allow you to get to the stage of theorizing, but it can't get you any further. This is why in the first few paragraphs of my original reply, I talked in very general terms about how faked data might differ from genuine data. These are supposed to give you places to go looking for anomalies, which you later investigate rigorously.
Looking for mistakes cheaters might make
Starting points can be things like testing to see if the numbers fit the conclusion too closely. If an experiment was done on several groups of test subjects, all of which are supposed to be identical, then you would expect the success rate in each group to be close to the overall average, but not too close. Some researchers who have made up their results had all group success rates equal to the average success rate to the nearest integer.
If you get someone to make up the results of 20 successive coin tosses, they deviate from statistical likelihood because they don't put eg. enough sequences of 5 heads in a row. People usually think things like this are less likely than they are. Look out for things which are 'too random' or 'too regular'.
Researchers into election fraud have had some success looking at the last two digits of numbers to see if double sequences like '11' or '22' occur less than they should, because humans who make up 'random' numbers tend to avoid them. This applies in the specific case where you have enough digits that the trailing digits should be uniform, but that no rounding should have been applied. This test wouldn't have detected the Chinese GDP rounding, or manipulations where leading digits are adjusted.
The mathematician Borel weighed the loaf of bread that his baker gave him each day and decided that the average was too far below the standard weight of a loaf to support the hypothesis that the baker wasn't making underweight bread. He confronted the baker, who promised that he would make the loaves heavier. After that Borel continued to weigh his bread. The average weight was now high enough, but he studied the distribution of weights and realized it corresponded to that which you would get if you always took the maximum of several observations of a normal distribution. He concluded that the baker always gave him the biggest loaf from those on the rack, but that the average was still below spec.
This is a classic illustration of how someone might falsify results - by taking the best result from several runs. In order to reason about the distributions, it was first necessary to understand how this method of cheating works.
Or suppose that someone had a bunch of results but threw out those which they didn't like. Has this introduced unlikely biases in the selection of the original test subjects? Eg if patients are supposed to be chosen at random but there are fewer old people than one might expect. In general if any data were rejected you should test for dependence between rejection and other variables.
Sometimes real data has a particular bias or noise which is lost in faked data. In the Simonsohn paper cited below, he looked at a psychological study where subjects were asked to say how much they would pay for a T-shirt. Unlike other, genuine studies, the results didn't cluster around multiples of $5.
Another thing which can be hard to look for but which is very damning is to figure out what the results might be if no effect was present and see if eg a single digit has been changed, or a round number has been added.
Sometime people genuinely do introduce biases unconsciously because they believe in their theories or want to succeed. This could mean that they make very small adjustments which can have a large cumulative effect, such as rounding up numbers which should be rounded down.
Trusting your 'feelings'
The other thing you need to try, is to get a 'feeling' for something dodgy, outside of the actual numbers. Again, all this does is give you a place where you try and build a proper statistical hypothesis and then test it against the data.
A mathematics professor once said to me that you can spot false proofs by two things: either the work becomes very complicated at the point where it is wrong, or the wrong step is skipped over as obvious. Not quite the same situation I know, but very complicated data handling procedures could be designed to be difficult to replicate (or could be the point where the researcher manipulated data until she got what she wanted). Saying something like 'cleaning' or 'normalization' without explaining exactly what was done could also be a red flag.
If there's a very very standard source of data of a particular type and someone didn't use it, or used it but not in its original form, why not? People often give a citation justifying some supposedly straightforward manipulation they perform on the data to clean it or get it in a more convenient form. Usually but not always, this reference should be to a standard textbook on stats or experiment design, or to some paper which everyone in the field knows. If it's to something extremely obscure is this justified by the obscurity of the topic? Does the cited work actually say what they claim it does?
How to proceed
I have tried to promote the general skill of trying to understand how people fake things, why and how they mix the truth with fabrication (or sometimes are subject to unconscious bias), and what constitutes strong evidence of anomaly. Looking at case studies, of which Simonsohn's paper is a great example, can help. Stephen Jay Gould's famous book 'The Mismeasure of Man', on the face of it a political tract critical of biological determinism, is also a collection of many case studies of deliberately or accidentally biased scientific work.
If you think that something is fishy, but you don't have the analytical tools you need to prove it, then you need to do research into specific statistical tests that apply to those cases. Among academics, most stats isn't done to detect fraud, and even if you have good quantitative skills you might not have this knowledge. The example of Borel is a good one in that many of us don't know offhand what the distribution of the 'biggest loaf to hand' should be, given some reasonable assumptions for the distribution of the loaf sizes.
However, as a researcher you should definitely have the skills to go and find this out from a book. Asking a statistician is a very important technique which may or may not be a last resort, depending on how friendly your statistician is.