In Cairo, stones and petrol bombs were thrown at the headquarters of the ruling Muslim Brotherhood.
Protest leaders have called on demonstrators to stay on the streets until Mr Morsi resigns.
They accuse the country’s first Islamist president of failing to tackle economic and security problems since taking power a year ago.
A presidential spokesman urged the protesters to respect the democratic process while Morsi supporters massed in the Cairo suburb of Nasr City.
One clear achievement of the protesters has been to get so many people out on to the streets, BBC’s Middle East editor Jeremy Bowen reports from Cairo.
The question they face now is how to fashion a political strategy that can rival the organisation of the Muslim Brotherhood, he says.
Another big question, he adds, is what the army will do. The minister of defence has warned that the military may intervene if Egypt becomes ungovernable.
Society split
One man was killed and at least 24 injured in Beni Suef, 115km (71 miles) south of Cairo, security sources said.
According to a report on the Ahram news website, Morsi supporters attacked an opposition rally and unidentified gunmen opened fire. The report could not be confirmed independently.
In the southern city of Assiut, three people died and several were injured when shots were fired at protesters, reportedly by attackers on a motorcycle.
According to the health ministry, 253 people were injured across the country during Sunday’s protests.
In Cairo, unidentified persons attacked the Muslim Brotherhood headquarters, which had been fortified with sandbags earlier. Staff inside said they had not broken in, Reuters news agency reports.
Demonstrations were also held in Alexandria, Port Said, Suez and other cities.
Presidential spokesman Ihab Fahmi called on all Egyptians to “unite and listen to the sound of wisdom.”
“Political diversity necessitates on all parties to abide by the democratic process,” he said.
He told reporters that the presidency was open to a “real and serious national dialogue”.
As darkness fell, the opposition National Salvation Front released what it called “Revolution Statement 1”, calling on protesters across Egypt to “maintain their peaceful [rallies] in all the squares and streets and villages and hamlets of the country… until the last of this dictatorial regime falls”.
The National Salvation Front is among liberal and secular opposition groups which have endorsed a petition organised by the grassroots movement Tamarod (Rebellion), which calls for a snap election. Opposition activists say more than 22 million people have signed it.
There was also some evidence of anti-American and anti-Israeli feeling among the protesters, with one flag portraying President Morsi inside a Star of David.
The BBC’s Aleem Maqbool in Cairo says many ordinary Egyptians – angered by Mr Morsi’s political and economic policies – are taking part in the rally in Tahrir Square.
At one point, army helicopters flew over Tahrir Square and dropped Egyptian flags, to cheers from the anti-Morsi crowd, he says.
Mr Morsi, who hails from the Muslim Brotherhood, became president on 30 June 2012, after winning an election considered free and fair.
His first year as president has been marred by constant political unrest and a sinking economy.
Analysis by Aleem Maqbool – BBC News, Cairo:
Many of those on the streets feel betrayed by a president who they say has been uninterested in uniting Egyptians despite once promising to be inclusive. They feel that he has instead governed purely in the interests of his own party.
For his part, the president insists he has invited opposition groups to enter into dialogue but that they have not co-operated. His supporters say that whatever the considerable problems Egypt is facing, Mohammed Morsi must see out his full term in office for the sake of stability.
Some pro-Morsi demonstrators have decided to stage their own sit-ins, an there are fears of clashes between opposing groups.
The coming hours, and the way the situation is handled by the authorities, are likely to give us clear indications about whether or not Egypt’s president can survive in office as he insists he will.
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