During the reign of Elizabeth I, Catholics were not given an easy rest. During the Elizabethan period, it was easy to be prosecuted for the religion being practiced other than the main Protestant faith that Elizabeth decreed.
So how can we tell if Shakespeare was a Catholic or not? The play Hamlet depicts the slain king suffering in purgatory, who tells Hamlet to avenge his death. Purgatory was a Catholic concept, with a soul being trapped in the region between heaven and hell, unable to go to heaven especially for "venial faults." Another example would be in Richard II, where on the stage the king confesses to use prayer with beads, another form of worship that met with hostility in England. Both forms of evidence were the basis of a retired Lawrence high school teacher turned KU graduate who looked into Shakespeare's faith.
Although there is a great possibility and speculation that Shakespeare was Catholic, the idea is not concrete. Not much is known about his life, for few records exist that depict how he lived. Shakespeare regardless wrote plays that are quite ambigous in nature so it would not have mattered to a Protestant, Catholic or a person of any other faith if they enjoyed the play. Shakespeare has not preached strong values in his time, he was only interested in creating entertaining dramas which persist to this day.