Much of the tension has focused on the al-Aqsa mosque compound, a hugely important site in Islam and Judaism.
The militant Palestinian Islamist movement Hamas had called for a "day of rage" over al-Aqsa.
Palestinians have also been angered by Israeli plans that could allow police to open fire on stone-throwers.
An Israeli motorist died earlier in the week in an accident apparently caused by a rock-throwing attack in Jerusalem.
Several security incidents were reported on Friday:
The al-Aqsa mosque compound itself was reported to be quiet on Friday. Israeli police set up checkpoints outside the site, limiting the age of worshippers to 40 and above for men.
The compound - known to Jews as the Temple Mount and to Muslims as al-Haram al-Sharif - is the holiest site in Judaism, and contains the al-Aqsa Mosque - the third holiest site in Islam.
Tensions have been running high in Jerusalem since Israel banned two Muslim groups which confront Jewish visitors to the compound.
Israel has deployed hundreds of extra police to Jerusalem after days of clashes between Palestinian protesters and security forces.