Are Catholics free to interpret Bible verses without the Church's approval?

Doctrine, not verses

The Church generally does not define how specific verses are to be taken. She defines doctrine, and a doctrinal definition may exclude some interpretations of particular verses. Ludwig Ott's Fundamentals of Catholic Dogma is a good guide here.

The seven passages

Only seven passages have had their sense partly — not fully — defined by the extraordinary magisterium, at the Council of Trent:

• "Born of water and Spirit" in John 3:5 includes baptism.
• "Do this in memory of me" in Luke 22:19 and 1 Corinthians 11:24: Jesus made the apostles priests.
• In Matthew 18:18 and John 20:22-23, Jesus gave the apostles the power to forgive sins, which not everyone shares.
• Romans 5:12 refers to the reality of original sin.
• The presbyters in James 5:14 are ordained, not simply older members of the community.