I assume that you don't mean "standing" in the legal sense (i.e. who has a right to sue over something).
The question of whether a building has structural integrity is an evidentiary issue. As a result, any competent evidence may be introduced to convince the finder of fact of the issue, and the finder of fact (in a civil matter) will resolve that issue based upon a preponderance of the evidence.
There are lots of ways that it could be proven (or disproven):
Some instances of lack of structural integrity would be so obvious that a mere authenticated photograph or video clip offered by a lay person would be sufficient to prove the issue.
A recent citation from a city inspector would argue strongly for a problem.
A recent issuance of a certificate of occupancy would argue strongly for a lack of a problem.
Any contractor with a license adequate to do that work would probably be competent to render an opinion.
Sometimes an architect would be competent to testify.
A licensed or certified building inspect might be competent to testify.
Structural integrity is usually the province of someone trained as a civil engineer with a structural engineering speciality. But often evaluating the condition of the soils would be considered a separate speciality to be evaluated by a soils engineer, often as a consultant whose work is used by the structural engineer to form an overall opinion.
Except in the case of a lack of structural integrity that it obvious to a lay person that can be proved with a simple photograph or video clip, or can be established with a citation or a certificate of occupancy, anyone presenting a factual opinion regarding the structural integrity of a building would have to be accepted by a court as an expert witness. A court's acceptance of someone as an expert witness is a process that would typically involve presentation of the qualifications of the expert in terms of education, licensure, experience, and history of publications in the area, as well as some defense of the validity of the discipline in which the expert witness is skilled as having a scientific basis and is relevant to the question at hand in this particular case.
In a fully litigated case, these issues are typically resolved in pre-trial motion practice and a pre-trial evidentiary hearing in the case of a dispute over whether the area in which the witness has expertise provides is a valid approach, as discipline, to answer the question in a scientifically reliable way. But this broad based attacks on an entire approach to addressing an issue don't usually come up in the case of the standard building and engineering professions.
Also, in a fully litigated case, the parties normally disclose an official report of the expert witnesses offered and make them available to be deposed prior to the trial. This requirement is often dispensed with, however, in litigation over preliminary matters.
The issue of what kind of expert testimony is competent to prove a disputed issue of fact is determined on a case by case basis. There is not a body of law that spells out in any real specificity what kind of evidence is or is not competent to prove the issue. The general rules of evidence apply and there might be (but probably isn't) case law under the rules of evidence for expert testimony that is on point to resolve your particular issue.
More often, there will be pronouncements in variously kinds of professional licensing statutes regarding what someone with a particular qualification is qualified to do, and those are evaluated with respect to a particular tendered witness in a manner that does not preclude someone else who might also have different qualifications that are also relevant to some aspect of the issue from also testifying to the factual issue presented.
Of course, that fact that an expert witness meets the minimum threshold to testify in court under the rules of evidence doesn't end the analysis. An expert witness is introduced in order to persuade the judge that the expert's opinion is correct. And, the more highly credentialed a witness is, the more relevant the credentials the witness holds, and the better the witness is at communicating the basis for the opinion of the witness to the judge or jury, the more likely it is that the finder of fact will give weight to the opinion of the witness and rule consistently with the opinion of the witness.
So, on one hand, a contractor or inspector would probably be less credible than a structural engineer working with a soils engineer. On the other hand, someone with great technical expertise who isn't a good communicator, or can't deftly respond to an opposing attorney's cross-examination question without falling into a "trap" laid by that attorney, could still be a poor expert witness.
Many technically skilled expert witnesses are more timid than necessary in formally giving opinions in court because they are used to discussing ideas in non-adversarial contexts where pointing out the limitations of your conclusions is welcomed and the colleague experts evaluating your opinion know not to give those limitations undue weight.