Did Jesus wear tefillin? Tefillin1 are boxes containing Torah verses, worn by Jews on the head and the upper arm, in accordance with instructions in Exodus 13:9, 13:16, and Deuteronomy 6:8, 11:18. There is a great deal of commentary in the Talmud and elsewhere about exactly how this should be done. 

Jesus says that the Pharisees were ostentatious in their wearing of tefillin - Matthew 23:5  (NIV) reads:


  Everything they do is done for people to see: they make their phylacteries1 wide and the tassels on their garments long. 


It seems from Mt 9:20, 14:36, Mk 6:56 that Jesus may have worn the same kind of fringes or tassels (tzitziyot), similarly commanded in the Law, albeit presumably his were not as long as those of the Pharisees.

Would Jesus also have worn tefillin, and why/why not? 
If he did (and his disciples too?) then when did Christians stop? If he didn't, then was this unusual or scandalous?

1 Tefillin (תפילין), or totafot (טוטפת), or phylacteries (φυλακτήρια)
In the first council of Jerusalem (In the Acts of the Apostles), we see an outline of the practices which the Jews did that they thought were reasonable to ask of the Gentiles. The wearing of Tefillin is not among them. 

In all likelihood after the destruction of Jerusalem and certainly after the Muslim incursions into the middle east, the number of culturally Jewish Christians was probably close to zero, and thus the practice of wearing Tefillin too would have disappeared.

Jesus may have worn Tefillin, we don't have a solid answer for that question. It is reasonable to assume that he didn't given his criticism of the practice. There are no records of him having worn them. It also would have been redundant.