The Utah State Law Library blog posted some helpful information on finding Utah court cases from about the last ten years:

Use the Utah State Courts’ web site to locate both Court of Appeals and Supreme Court opinions. Court of Appeals opinions are available from 1997 to present and Supreme Court opinions are available from September of 1996. Use the search box to do a keyword search, or browse opinions by name or by year of opinion.

Indexes, case files and registers that are much older are available for many Utah district courts, consult the District Court Records Guide to determine what is accessible at the Archives based on the county where the case was filed.