I suppose it might depend on local rules, but around here you’re generally discussing cases you’ve already included in your brief, so a mention of the style and who the court was is usually sufficient: “as your Honors said in Doe v. Roe…” or “as the 5th Circuit ruled in In Re Blackacre…”. If the Court wants a complete citation they’ll let you know.
ETA: I was assuming that by oral argument you meant appellate court. In making an argument in trial court, the judge may or may not be familiar with the case you’re citing and may not have had sufficient time to digest your brief (if any), so a more complete citation and explanation may be necessary.
Presumably it also depends on how well-known the case is? I mean, if you’re citing Miranda v. Arizona you probably don’t have to explain that it’s 384 U.S. 436.