* At least 3 protesters killed amid Cairo clashes
* Islamists opposed to Mursi's overthrow take to streets
* Military's handling of unrest watched closely from abroad (Adds more details, quotes)
At least three protesters were shot dead on Friday outside the Republican Guard barracks in Cairo where deposed President Mohamed Mursi is being held, security sources said, as angry Islamist supporters confronted troops across the country.
A crowd of several hundred people marched towards the barracks as part of what Mursi's Muslim Brotherhood movement has called a "Friday of Rage" to protest against his ouster.
Egypt's first freely elected president was toppled on Wednesday in what his Islamist supporters call a military coup.
Television footage showed Mursi supporters with what appeared to be shotgun pellet wounds.
The army denied blame. An army spokesman said troops did not open fire on the demonstrators and soldiers used only blank rounds and teargas to control the crowd. It was unclear whether security forces units other than army troops were also present.
Before the demonstrations, a military source said the armed forces would allow people to demonstrate where they wanted.
Continued violence will alarm the United States. Washington has so far avoided referring to the army's removal of Mursi as a "coup", a word that under U.S. law would require a halt to its $1.5 billion in annual aid. Mursi's opponents also say it was not a coup but an intervention to impose the "people's will".
Thousands of Islamists took to the streets of Alexandria and Assiut to join protests, opposing a planned interim government which is backed by their liberal opponents. Egypt's liberal coalition issued an "urgent call" for its supporters to take to the streets in response to Islamist protests.
In Damanhour, capital of the Beheira province in the Nile Delta, 21 people were wounded in violence between the factions.
Ehab el-Ghoneimy, manager of the Damanhour general hospital, said three people had been wounded with live bullets, others were wounded with birdshot, rocks, or had been hit with rods.
In the Suez city of Ismailia, soldiers fired into the air as Mursi supporters tried to break into the governor's office. The Islamists retreated and there were no casualties, security sources said.
Egypt has been in turmoil since the fall of Hosni Mubarak in the "Arab Spring" revolutions that swept the region in 2011.
FROM JOY TO ANGER
Several dozen people have been killed in the last month of unrest, during which huge rallies in Cairo and other cities called for Mursi's resignation amid anger over economic stagnation and perceptions of a Brotherhood power grab.
His overthrow on Wednesday was greeted with wild scenes of celebration involving millions of people, but also infuriated his supporters who fear a return to the suppression of Islamists they endured under generations of military rule.
Hoda Ghaneya, a leading female figure in the Brotherhood, said she and two of her sons accompanying her at a Cairo rally after Friday prayers were ready to sacrifice themselves to the cause, but added that they would not turn to violence.
"We will die not as a sacrifice for Mursi, but so the Egyptian people recover their freedom," she told Reuters near the Rabaa Adaweya mosque in a Cairo suburb that has been the centre of Islamist protests in the last few days.
Tens of thousands of people had gathered there following Friday's violent clashes.
In the Sinai Peninsula bordering Israel, gunmen fired rocket-propelled grenades at army checkpoints guarding an airport and rocketed a police station near the border with the Palestinian territory of Gaza. One soldier was killed and two wounded, a security source said.
An army spokesman said the army in the Sinai Peninsula was "on alert". He denied an earlier report by state-owned media Al-Ahram that a state of emergency had been imposed in the South Sinai and Suez provinces, which had caused a spike in oil prices from international markets on edge over the unrest.
How the army deals with the unrest on Friday and beyond will help determine future support for Cairo from the United States and other international powers.
A military source said: "We will continue to secure the places of protest with troops, and jets if necessary, to make sure the pro- and anti-Mursi demonstrators don't confront each other. We will let them demonstrate and go where they want."