Jesus: Sabbath in Nazareth
Since many of Jesus' Sabbath events and teachings are mentioned in Luke, we will follow Luke's chronology for most of them, then look at the others that Luke doesn't mention.
The first event involves controversy, but not regarding the Sabbath. The Sabbath is simply the occasion for Jesus' teaching. It is the teaching itself that people take issue with. In Luke 4:16, as his first recorded public appearance, he went to the synagogue in Nazareth, "as was his custom". These people knew him; he had been attending synagogue every Sabbath day here for decades. They gave him the honor of doing one of the public readings of scripture. After the Torah portion of the week was read, a reading from the prophets that was consistent with the weekly Torah reading would be given. Jesus is selected for this reading, and the prescribed passage is opened for him, which he reads.
After reading, he sits down. This is not sitting down back in the "pews". The congregation was probably standing. The teacher would sit down at the "bema" to teach (cp. Matt. 23:2). That was the position of authority. When Jesus sat down, everyone's attention was focussed on him because he was beginning to teach. His teaching consisted of the idea that he was the fulfillment of the prophet's words, but that people in his hometown would not accept him. He was too familiar to them to seem authoritative. This message made the people angry and they tried to harm him but were unable to do so.
In this incident the Sabbath seems to function primarily as a backdrop for Jesus' teaching. There is no disagreement concerning it. All parties are honoring it and using it as their day of worship.
A similar description occurs in Mark 6:2-6 and Matthew 13:53-58. In these accounts it happens later in his ministry and his disciples are present. But the particulars of the story are similar. The people in the Nazareth synagogue take offence at his teaching and miracles because he grew up among them.
Sabbath in Capernaum