The Syrian Observatory for Human Rights said three people, including a child, were killed as regime forces opened fire to disperse protests.
"Tens of thousands of people protested today in various areas of the country," Rami Abdel Rahman, head of the Britain-based group, told AFP.
He said one protester was killed in the village Daf al-Shok in Damascus province. Another died in the Sakhur district of northern Aleppo, Syria's second city, and the child was killed in the eastern city of Deir Ezzor.
Three members of the security forces and a deserter were also killed in other clashes across the country, the Syrian Observatory said.
At least 11 people died and 28 were wounded in the Damascus bomb blast which hit as worshippers were leaving weekly Muslim prayers at nearby Zein al-Abidin mosque in the central Midan district, state television said.
The report blamed "terrorists," the term used by President Bashar al-Assad's regime to refer to the armed opposition, and said civilians and security force members were among the casualties.
TV footage showed gruesome images, including a severed hand and leg.
The official SANA news agency reported the interior ministry as saying "it will not tolerate the armed terrorist groups and vowed to strike with an iron fist those who are terrorising citizens."
A separate blast hit an industrial zone of Damascus where there were no reports of casualties, but three security agents were wounded in an explosion in the coastal city of Banias, the Observatory said.
Assad's regime has repeatedly blamed "armed terrorist groups" for the violence, and for failing to abide by a putative ceasefire since April 12.
Neighbouring Lebanon on Friday intercepted a ship suspected of carrying weapons destined for Syria's rebels, a Lebanese security official said.
But the US State Department said Washington still believed the regime was responsible for "the bulk of the violations" of the ceasefire, warning it was ready to return to the UN Security Council for action on Syria.
UN chief Ban Ki-moon also said Damascus was in contravention of a six-point peace deal by keeping troops and heavy weapons in urban areas, and expressed alarm at reports of population centres being shelled.
EU foreign policy chief Catherine Ashton said she was "extremely concerned about the continued violence in Syria in violation of the ceasefire".
Ashton said it was "clear that the Syrian government is not fulfilling its obligations and is failing to meet its commitments to withdraw its troops and heavy weapons from population centres".
Rights group Amnesty International said on Friday it had received the names of 362 people reportedly killed in Syria since UN observers deployed last week to monitor the peace deal.
More than 9,000 people have died since a popular uprising erupted against Assad's regime in March 2011, the UN says, while non-governmental groups put the figure at more than 11,100.
Opposition figure Walid al-Bunni said the peace deal drawn up by UN-Arab League envoy Kofi Annan was likely to fail because it obliges Syria to allow free demonstrations.
"If the Annan plan, which provides for peaceful demonstrations, is applied, millions of Syrians will take to the streets and the regime will fall," he told AFP in Cairo.
Syria's exiled Muslim Brotherhood urged Ban to acknowledge that Damascus had failed to honour the peace plan and to suspend its UN membership until a transitional government representing the Syrian people is formed.
"We ask Ban Ki-moon to announce that Assad's government has failed to honour the peace plan and to declare the plan finished... at a time when dozens of innocent people are dying," the group said in a statement.
Ban himself on Friday, during a visit to New Delhi, reiterated his alarm at the continuing violence in Syria.
"The continued repression of the civilian population is totally unacceptable. It must stop immediately. The government of Syria must live up to its promises to the world," he said.
The shaky truce is to be monitored by 300 UN observers due in the coming weeks. A small advance team is already in Syria, and the numbers will be doubled to 30 by Monday, according to a UN official.
The UN on Friday appointed Norway's Major General Robert Mood to head the monitoring force. Mood, 54, negotiated conditions for the advance team with Damascus.
Western nations have expressed strong doubts that the observers will be able to work, however, and the United States has already warned it may not renew the mission's initial three-month mandate.
NATO chief Anders Fogh Rasmussen said Syria was not respecting the Annan peace plan, but added: "We have no intention to intervene in Syria. We believe the right way forward is to ensure a political, peaceful solution."
Meanwhile the UN refugee agency's Syria Regional Refugee Response web page says there are more than 65,000 Syrian refugees in neighbouring countries.
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