It depends
australia
“Standard of review” is not a term that is used in Australia, instead, the governing legislation would determine if an appeal were to be strict, a rehearing, or a rehearing ‘de novo’. Similarly, whether there is a right to appeal, what can be appealed, what the court may consider, and what powers they have are defined by the statute.
Note that this is the situation for appeals: merits review of administrative decisions and common law judicial review are different and are addressed later. In casual usage, the word “appeal” might be used for any of the three but formally they are different.
For this question, ‘de novo’ hearings are the easiest to deal with - the trial judge’s reasoning is immaterial as the appellate courts retries the case from the beginning.
In a strict appeal the appellate court only considers the law at the time of the original trial and the evidence that was before it. It is also limited to only those matters that the parties raise in the appeal - there might be a glaring and obvious error in the decision but if its not raised in the appeal, its not for the court to go looking for it. The standard that is applied is whatever the statute says or, if it is silent, whether the decision is correct or not on the matters in dispute in the appeal.
In a rehearing, the appellate court may consider subsequent changes in the law and may allow fresh evidence on the facts. This may mean that the decision was correct at the time but is incorrect now. Again, the statute dictates the standard to be applied with the default being correctness.
Merits review of administrative decisions are not done by the courts but by internal or external administrative review bodies. For example, each state and territory has a civil and administrative tribunal which are known by their acronyms - NCAT for NSW, VCAT for Victoria, QCAT for Queensland etc. A merits review looks at the correctness of the decision as well as considering if the decision was the “best” in the circumstances - it may overturn a correct decision for a better correct decision.
There were traditionally four grounds for applying for judicial review of a decision:
the decision was ultra vires (it was beyond the legal power or authority of the decision-maker to make the decision)
the decision was made in breach of the rules of procedural fairness (for example, if the decision-maker did not give the person concerned a fair hearing)
in the case of a court or tribunal, there was a jurisdictional error (the court or tribunal mistakenly decided that it had (or did not have) jurisdiction over a particular matter)
in the case of a court or tribunal, there was an error of law on the face of the record (there was a mistake of some kind in the record of the court’s decision, not in its reasons for decision).
Again, legislation can limit the scope of these. For example, the arbitration acts severely restrict the grounds for judicial review of an arbitration, among other things, an error of law on the face of the record is not reviewable.