Activists said there were some clashes on the southern edge of eastern Ghouta along with two airstrikes just hours after the resolution was unanimously adopted on Saturday night.
On Sunday aerial bombing had stopped and the Syrian Observatory for Human Rights said there had been no deaths since the resolution was passed following a week of airstrikes.
Ghouta-based opposition activist Anas al-Dimashqi said the night was calm but warplanes and drones were flying over rebel-held areas and several explosions were heard.
Dr Sakhr al-Dimashqi, a surgeon at a clinic in Ghouta, said several shells hit some towns in the suburbs and six wounded people were treated at the clinic.
But he said: “The shelling today is not as intense as over the past week.”
The observatory said the death toll had hit almost 500, including 165 children and 64 women.
Two of the largest and most powerful rebel factions in Ghouta, Failaq al-Rahman and Army of Islam, said they would abide by the 30-day ceasefire unless they are forced to fire in self-defence. Both called for the “immediate delivery” of emergency aid.
The resolution excludes the Islamic State and al Qaeda -linked fighters and Ghouta is also home to a few hundred members of the al Qaeda-linked Levant Liberation Committee.
Residents in Damascus reported more traffic on the streets compared to previous days and most schools and universities were open.
They said some private schools were still closed, especially those close to the front lines with Ghouta.
Saturday’s United Nations vote came after being delayed on Friday when Russia’s UN Ambassador Vassily Nebenzia had repeatedly called an immediate ceasefire unrealistic.
In a bid to get Russian support, sponsors Kuwait and Sweden amended the draft resolution late on Friday to drop a demand that the ceasefire take effect 72 hours after the resolution’s adoption.
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After two hours of extra negotiations, the Security Council unanimously approved the resolution demanding a 30-day ceasefire across Syria “without delay” to deliver humanitarian aid to millions and evacuate the seriously ill and wounded.
Russia has been a main backer of Syrian President Bashar Assad since the country’s conflict began seven years ago. In 2015, Moscow joined the war on Assad’s side tipping the balance of power in his favour.