ISIS, the Sunni terror movement, claimed responsibility for both attacks, but CNN cannot independently verify its assertion.
The attacks are the latest strikes in a country on edge over violence during the holy month of Ramadan and the intensified war against ISIS. Operations are underway in both Syria
One of Friday's strikes killed and wounded civilians in an outdoor market in the town of al-Musayab. At least 22 people died and more than 30 others were wounded, security officials in Iraq told CNN.
The majority Shiite town is about 75 kilometers, or more than 46 miles, south of Baghdad in Babil province.
Women and children were among the victims, security officials said.
In Karbala, a suicide bomber injured at least 10 people. The city is about 27 kilometers, or 17 miles, west of al-Musayab.
Karbala, a holy and historic city in Shiite Islam, is in Karbala province.
Amaq New Agency, the media voice for ISIS, claimed responsibility for the terror group in both attacks in separate statements Friday on social media.
There have long been sharp tensions and sectarian violence between Sunnis and Shiites
, adherents of the largest branches in Islam.
, a month of prayer and fasting that began in late May, is also a time jihadist groups ramp up attacks.
One bomb in the early hours of May 30 exploded outside an ice cream shop
in the heart of Baghdad, targeting young families enjoying a late-night Ramadan snack.
Melted ice cream mixed with blood as the dead and injured lay on the ground, witnesses said.
Ten people were killed and 40 wounded.
Hours after that blast, a second bomb went off as elderly residents lined up to wait for their pension payments. At least 12 people died and 45 were injured.
Last Ramadan, on July 3, ISIS claimed responsibility for an attack in Baghdad's Karrada neighborhood
that left more than 380 people dead.