As far as I know, none. Romans didn't recognize "subjects." If he were a citizen (as Paul was) he could have appealed to the emperor, but without citizenship Roman law didn't give a flying fuck. If he were upper class, he might have had some access to imperial government through patronage networks, but what patron is a carpenter from Galilee going to have?

Roman provinces are, I think, technically theaters of war, even if there is not an active conflict or an army present. The senate can issue orders to governors which sometimes governors felt compelled to accommodate (as Cicero did in his tenure in Cilicia), and the senate could... well impeach isn't quite the right word, but Cato thought Caesar was acting out of line in his tenure of Cisapline Gaul, and wanted to bring charges. But in practice? What is the senate going to do? The governor's hundreds of miles away and probably has troops. He acted pretty autonomously.

As a Roman subject, not many. Truly "Roman" law was reserved for citizens, which Jesus was not.

Very broadly, any given area of the Empire (I think 'county' might be the equivalent, but the Latin escapes me so thoroughly that I can't manage to Google it) had its own set of traditional laws often handed down from before the area was under Roman influence and, for the most part, Rome was happy to leave those laws stand. So Jesus would have largely been subject to whatever the laws of Judea (or maybe some finer-grained sub-section of Judea) were. Jewish law, in other words.

Rome was generally pretty hands-off, so long as the taxes got paid and no one proclaimed themselves Augustus. Or started rebellions, in the case of the Jews.

There are exceptions, obviously, most of them involving rights Jesus did not have as a peregrius (that one I could Google), a non-citizen resident of a province. For example, if Jesus was ever involved in a lawsuit with a Roman citizen, then Roman law superseded local law. Citizens were also protected by a do-not-torture-me law, a right non-citizens did not enjoy. If Rome ever needed to torture Jesus for any reason (including reasons such as "we just wanted to torture someone") then Jesus could be tortured without recourse. Killed too, if the governor said so.

For the most part though, Rome wasn't interested in individual peregrius. In the Bible, Jesus is not arrested by the Romans, he is arrested by the local Jewish authority. He is then tried by the Jewish authority. The Romans only enters the story after the Jews decide they want Christ executed. Apparently capital crimes were the responsibility of Roman prefects (I'm basing this entirely on the Biblical account). At no point is the Roman government particularly interested in Christ one way or the other. Pilate even tries to foist the case onto Herod, who immediately foists it back (with an implied yawn).

And, not really asked, but answered anyway, non-citizens could petition to be made citizens on an individual basis. It happened with fair regularity. Then Caracalla made everyone a citizen while simultaneously ignoring most of the rights of citizens.

Was Jesus a roman subject ? As he was born in Galilee, he was the subject of the King of Galilee, Herod Antipas.

That at least is why the gospel of Luke says Pilate decided that Jesus was under the jurisdiction of Herod and tried to make him handle the matter.

There are reasons to believe that this story is not historical, and was added later, but the point remain.

In the end Jesus was judged and condemned by Pilate because of the actions he made in Jerusalem (that was under the roman jurisdiction), but technically he was not a roman subject but a subject of a roman client-king.