'Reasonable investigation' is a legal test and it's difficult to easily identify a list of components that would show it was 'reasonable'. However, if I were alleging that a fair procedure had not been followed, I would be looking at every stage for school policies to have been followed (which should be at least in line with the ACAS Code of Practice, as @nomad said); evidence of the decision-making process that gave rise to the formation of allegations; who made that decision and why; what alternatives were considered to sending you home, why a decision was made to do so and who made it. Did they seek HR advice or act unilaterally? At every stage of the investigation I expect to see fair evidence-gathering procedures and evidence of an evaluation of the evidence, to justify whatever next step is taken. As for proving a witch-hunt, there is always an evidence trail, even if it's not immediately apparent. It may or may not be easily available for an internal investigation but could be requested. Failure to provide it may be further evidence of unfair procedure. If it got as far as litigation, disclosure of this evidence can be forced if necessary via an Application for an Order. This is something that could be discussed in further detail in camera.