(2:1b-2) "There came wise men from the east"
The Revised English Bible translates this as, "After his birth astrologers from the east arrived in Jerusalem...." If this translation is correct, then Matthew's gospel seems to approve of astrology.
Strange Flesh: What about the magi?
Queer Epiphany: Three kings or three queens?

(2:5-6) "In Bethlehem of Judaea"

Matthew claims that Jesus's birth in Bethlehem fulfils a prophecy from Micah 5:2. But this is unlikely since:

• "Bethlehem Ephratah" in Micah 5:2 refers not to a town, but to a clan: the clan of Bethlehem, who was the son of Caleb's second wife, Ephrathah (1 Chr.2:18, 2:50-52, 4:4).

• The prophecy (if that is what it is) does not refer to the Messiah, but rather to a military leader, as can be seen from verse 5:6. This leader is supposed to defeat the Assyrians, which, of course, Jesus never did.

Whatever the correct interpretation of Micah 5:2 may be, the author of Matthew thought that it required the Messiah to be born in Bethlehem. So he found a way to make that happen, as did the author of Luke, probably for the same reason. They achieved it in different ways, however. Matthew's gospel has Joseph and Mary living in a house in Bethlehem when Mary becomes pregnant; Luke has them living in Nazareth and traveling Bethlehem because of a census. Neither Mark nor John mention anything about a birth in Bethlehem, although the author of the gospel of John was aware that some expected the Messiah to be born there (John 7:41-42).

(2:9) "The star, which they saw in the east, went before them, till it came and stood over where the young child was."
If the star "went before them," leading them to Bethlehem, then it couldn't have been a star or any other astronomical object or event. But Matthew couldn't have known that. Everyone at the time thought that stars were just little points of light a short distance above the earth. It'd be no problem to have one hover above a particular place for a while.

(2:11) "And when they were come into the house"
Was Jesus born in a house or a manger?
Where did Joseph and Mary live before the birth of Jesus?

(2:14) "He took the young child and his mother by night, and departed into Egypt:" According to Matthew, Mary, and the baby Jesus left for Egypt soon after Jesus' birth, yet Luke (2:39) says they went directly to Nazareth.
Did Jesus, Mary, and Joseph go to Egypt or Nazareth?

(2:15) "Out of Egypt I have called my son."
Matthew claims that the flight of Jesus' family to Egypt is a fulfillment of Hosea 11:1. But Hosea 11:1 is not a prophecy at all, as is clear when the entire verse is quoted ("When Israel was a child, then I loved him, and called my son out of Egypt."). It is a reference to the Hebrew exodus from Egypt and has nothing to do with Jesus. Matthew tries to hide this fact by quoting only the last part of the verse.

(2:16) "Herod ... slew all the children that were in Bethlehem ... from two years old and under." Herod kills all boys in and around Bethlehem that are two years old and under. Such a massacre would certainly have been noted by contemporary historians. Yet not even Josephus, who documented Herod's life in detail, mentioned this event.

(2:17) "Then was fulfilled that which was spoken by Jeremy the prophet"
Matthew quotes Jeremiah 31:15, claiming that it was a prophecy of King Herod's alleged slaughter of the children in and around Bethlehem after the birth of Jesus. But this verse refers to the Babylonian captivity, as is clear by reading the next two verses (16 and 17), and, thus, has nothing to do with Herod's massacre.