The Wald, Likelihood Ratio and Lagrange Multiplier tests in the context of maximum likelihood estimation are asymptotically equivalent. However, for small samples, they tend to diverge quite a bit, and in some cases they result in different conclusions.
How can they be ranked according to how likely they are to reject the null? What to do when the tests have conflicting answers? Can you just pick the one which gives the answer you want or is there a "rule" or "guideline" as to how to proceed?
I do not know the literature in the area well enough to offer a direct response. However, it seems to me that if the three tests differ then that is an indication that you need further research/data collection in order to definitively answer your question.
You may also want to look at this Google Scholar search
Update in response to your comment:
If collecting additional data is not possible then there is one workaround. Do a simulation which mirrors your data structure, sample size and your proposed model. You can set the parameters to some pre-specified values. Estimate the model using the data generated and then check which one of the three tests points you to the right model. Such a simulation would offer some guidance as to which test to use for your real data. Does that make sense?
I won't give a definitive answer in terms of ranking the three. Build 95% CIs around your parameters based on each, and if they're radically different, then your first step should be to dig deeper. Transform your data (though the LR will be invariant), regularize your likelihood, etc. In a pinch though, I would probably opt for the LR test and associated CI. A rough argument follows.
The LR is invariant under the choice of parametrization (e.g. T versus logit(T)). The Wald statistic assumes normality of (T - T0)/SE(T). If this fails, your CI is bad. The nice thing about the LR is that you don't need to find a transform f(T) to satisfy normality. The 95% CI based on T will be the same. Also, if your likelihood isn't quadratic, the Wald 95% CI, which is symmetric, can be kooky since it may prefer values with lower likelihood to those with higher likelihood.
Another way to think about the LR is that it's using more information, loosely speaking, from the likelihood function. The Wald is based on the MLE and the curvature of the likelihood at null. The Score is based on the slope at null and curvature at null. The LR evaluates the likelihood under the null, and the likelihood under the union of the null and alternative, and combines the two. If you're forced to pick one, this may be intuitively satisfying for picking the LR.
Keep in mind that there are other reasons, such as convenience or computational, to opt for the Wald or Score. The Wald is the simplest and, given a multivariate parameter, if you're testing for setting many individual ones to 0, there are convenient ways to approximate the likelihood. Or if you want to add a variable at a time from some set, you may not want to maximize the likelihood for each new model, and the implementation of Score tests offers some convenience here. The Wald and Score become attractive as your models and likelihood become unattractive. (But I don't think this is what you were questioning, since you have all three available ...)
