The US military said an American soldier was killed in the blast.
Giving a total for the deaths in the city, home to some six million people, the source added: "Nineteen were killed and 40 wounded in a suicide attack on Aden Square. The other victims were - four killed in an attack in Baya, and four in an attack on Al-Nida mosque."
The official, who asked not to be named, added that 109 people were wounded.
In the district of Baya, a suicide bomber, also on a bicycle, rode into a Sunni funeral tent, killing four and wounding 37, medics said.
"A black-skinned man arrived on a bicycle at around 12:45 (0945 GMT) and rode into the tent before exploding," Mohammed Khalil Ibrahim, a relative of the deceased, told AFP.
A mortar attack close to the Al-Nida mosque in the northern Baghdad Sunni district of Adhamiya also killed four, the government source said.
Two people died in Baquba, north-east of the capital, in an apparent attempt to assassinate an Iraqi general, security sources said, while two Iraqis were killed by a shell in Samarra, and two policemen died from an improvised bomb north of the city.
The Baquba attack was claimed by militants loyal to Iraq's al-Qaeda front man, Abu Musab al-Zarqawi, in an Internet statement whose authenticity could not be verified.
In the northern oil city of Kirkuk, a Sunni sheikh linked to the Kurdistan Democratic Party, one of the two main Kurdish parties, was shot dead.
Near Balad, south of Samarra, the corpses of five soldiers were found overnight riddled with bullet and with their hands tied.
Meanwhile police said Iraqi security forces had arrested Haidar Abu al-Buwari, the alleged commander of an insurgent cell close to Zarqawi in Diyala province, whose capital is Baquba.
"He is one of the Mujahideen princes who works with Zarqawi in the position of cell leader," a police spokesman said.
Meanwhile, horse-trading for the top jobs in a new government has already started, but little appears to have been decided, and choosing a presidential council and a new cabinet could take days if not weeks of wrangling.
Winning parties have promised to involve Sunni Arabs in the assembly's work of drafting a new constitution to avoid fanning the former elite's sense of grievance and boosting support for the insurgents.
In the latest hostage crisis, Indonesian President Susilo Bambang Yudhoyono and the mother of one of two Indonesian journalists held captive in Iraq made emotional appeals for their safe release.
In Rome, up to half a million peace demonstrators marched to demand the release of Italian journalist Giulian Sgrena, held hostage in Iraq since February 4.