In the deadliest attack, a suicide bomber struck at a market in Baghdad's Dora district, killing 12 people, including three US soldiers.
The Kirkuk blast targeted members of the Awakening Council, a US-allied Sunni militia, killing eight people.
A third bombing inside a Baghdad police station killed three recruits.
Concerns over violence
The attacks come a day after at least 40 people were killed in a car bomb attack in north-western Baghdad.
Both the Iraqi government and the US say recent attacks are isolated incidents that do not undermine security gains. They say the attacks are not as sophisticated as they once were.
April was the bloodiest month in Iraq this year, with a 40% rise in the number of people killed over March - but the toll for May had been lower.
However the BBC's Natalia Antelava, in Baghdad, says many people in the capital feel the situation is deteriorating and could get worse once US troops withdraw from Iraqi cities at the end of June.
The US military said it could not immediately confirm Iraqi police reports of the deaths of three US soldiers in the attack in Dora, a district that was one of the city's most dangerous areas until a year ago.
The Kirkuk blast earlier on Thursday targeted members of the so-called Awakening Councils - a movement of former militias and insurgents who have turned against al-Qaeda in Iraq and allied themselves with the US and Iraqi militaries.
The attacker, dressed in the group's uniform, joined a group of men waiting to receive their pay and then detonated explosives, officials said.
The attack at Baghdad's Al-Mamoun police station, in the west of the city, claimed the lives of three policemen and injured 19 people, including eight civilians.
Police said the bomb was planted inside a garbage container - the first time an explosive had gone off inside a police station.