Supporters and opponents of President Mursi's decree held rival demonstrations across Egypt on Friday
Protesters in Egypt have attacked Muslim Brotherhood offices in several cities, according to state TV.
They were protesting against President Mohammed Mursi's decree granting himself sweeping new powers.
The decree states the president's decisions cannot be revoked by any authority - including the judiciary.
Rival rallies are being held nationwide, with the president telling one he was leading Egypt on a path to "freedom and democracy'".
'Mursi is Mubarak'
Offices of the president's Muslim Brotherhood party have reportedly been attacked in the cities of Port Said and Ismailia. Earlier reports of an attack on the office in Suez were denied.
Clashes between rival demonstrations have been reported in Alexandria. Anti-Muslim Brotherhood protesters in the Mediterranean city stormed the offices of the group's Freedom and Justice Party, throwing out books and chairs.
In Cairo, thousands of opponents of Mr Mursi heeded calls to protest against the decree in Tahrir Square.
Chants of "Mursi is Mubarak... revolution everywhere" rang out.
Thousands of supporters of the president gathered outside the presidential palace, some holding up photographs of Mr Mursi.
He later appeared in front of the supporters there, telling them he was the guardian of political, economic and social stability.
Vowing to protect the rights of the opposition, Mr Mursi said: "There is a great future for this nation."
In an earlier address at a mosque in a Cairo suburb on Friday, Mr Mursi said: "We are, God willing, moving forward, and no-one stands in our way.
"I fulfil my duties to please God and the nation and I take decisions after consulting with everyone."
According to Egypt's state-run news agency, Mena, three people were injured in violence on Friday in Cairo's central Mohammed Mahmoud street.
In a joint news conference on Thursday, Sameh Ashour, head of a lawyers association, and key opposition figures Mohamed ElBaradei and Amr Moussa accused Mr Mursi of "monopolising all three branches of government" and overseeing "the total execution of the independence of the judiciary".
"We are calling on all Egyptians to protest in all of Egypt's squares on Friday," they said.
Mr ElBaradei, a Nobel Peace Prize winner, wrote on his Twitter account that the president had "appointed himself Egypt's new pharaoh. A major blow to the revolution that could have dire consequences".
Critics say the decree is a "coup against legitimacy".
The BBC's Jon Leyne in Cairo says that after brokering a ceasefire between Israel and Hamas over the Gaza crisis, Mr Mursi has won a new international prestige, and must believe this will help him weather the domestic storm.
The new decree bans challenges to his decrees, laws and decisions, and says no court can dissolve the constituent assembly, which is drawing up a new constitution.
It also opens the way for a retrial of people convicted of killings during Egypt's 2011 uprising which toppled President Hosni Mubarak.
Mr Mursi has sacked chief prosecutor Abdel Maguid Mahmoud and ordered the retrial of people accused of attacking protesters in the uprising, which could lead to the retrial of Mubarak himself, who is serving a life sentence.
Mr Mahmoud's acquittal of officers accused of involvement in attacks on protesters led to violent clashes in Tahrir Square in October, when supporters and opponents of Mr Mursi clashed.
The president had tried to remove Mr Mahmoud from his post by appointing him envoy to the Vatican. But Mr Mahmoud returned to work, escorted by judges and lawyers.
New prosecutor Talaat Ibrahim is tasked with re-examining all the investigations led by Mr Mahmoud into the deaths of protesters, and re-trying people already acquitted in the case.
The declaration also gives the 100-member constituent assembly two additional months to draft a new constitution, to replace the one suspended after Mr Mubarak was overthrown.
The rewrite of the constitution, which was meant to be finished by December, has been plagued by lawsuits questioning the make-up of the constituent assembly.
Once completed, the document is due to be put to a referendum. If it is approved, legislative elections will be held two months later.