The largest blast, which Islamic State said it was behind, came from a parked car bomb in the Saydiya district of southern Baghdad that killed 11 and wounded 30, the sources said.
At least a few of the casualties were pilgrims passing through the area on their way to the shrine of Imam Moussa al-Kadhim, a great-grandson of Prophet Mohammad who died in the 8th century.
Explosives planted on the ground in Tarmiya, 25 km (15 miles) north of Baghdad, killed two and wounded six, while a roadside bomb in Khalisa, a town 30 km (20 miles) south of the city, left one dead and two wounded. There was no immediate claim of responsibility for the smaller attacks.
Islamic State militants fighting Iraqi forces in the north and west regularly target security personnel and Shi'ite civilians whom they consider apostates.
The group said in an online statement distributed by supporters that a suicide bomber had targeted pilgrims in the Dora neighborhood adjacent to Saydiya. It said the attack was part of an offensive launched recently in apparent revenge for the killing of a senior leader.
Islamic State's al Qaeda predecessor was blamed in the past for such attacks on Shi'ite pilgrims, including blasts in 2012 that left 70 people dead nationwide.
Security has gradually improved in Baghdad, which was the target of daily bombings a decade ago, but there has been a string of blasts in recent days, including a suicide attack on Saturday that killed at least 19 people.
May 2's blasts come as Iraq struggles to emerge from a political crisis over reforming its governing system which saw protesters hold an unprecedented sit-in over the weekend in Baghdad's heavily-fortified Green Zone.