May a husband or wife check their spouse's phone without permission?

In general, spying on a spouse's phone without permission is prohibited because the Qur'an forbids tajassus (searching out people's faults).
Verse 49:12 commands avoiding much suspicion and not spying. Marriage is built on trust; secret checking usually breeds more suspicion and quarrels. It is better for spouses to agree on transparency — for example sharing passwords voluntarily. If there are strong signs of danger (for example a threat to the children or to safety), refer to the authorities rather than acting alone. Evidence obtained wrongfully may also be problematic in court. Discuss your concerns directly and get counselling if trust has already cracked.