BAGHDAD — At least three dozen people were killed in a string of attacks Monday in Iraq, the police said, signaling a bloody end to a year of rising ethnic, political and sectarian troubles that showed little sign of easing.

While violence has been less frequent and widespread in Iraq than it was at the height of the insurgency, rebels have continued to strike at a variety of targets, with the apparent intent of exploiting sectarian and cultural divides.

In the last two months of 2012 alone, bombers attacked a political party, a military headquarters, a busy marketplace and a university campus. More than two dozen people were killed in one bomb attack at a military base in Taji on Nov. 6. At least 30 were cut down across the country Dec. 17 in a wave of shootings and suicide and car bombings.

Iraq has been unsettled recently by the absence of President Jalal Talabani, a Kurd who was an important calming influence. He had a stroke and was taken to Germany. War in neighboring Syria has also had a destabilizing effect, energizing the Sunni militias that fought in the anti-U.S. insurgency.

As 2012 drew to a close Monday, police officials provided yet another violent accounting of bloodshed, with at least 36 civilians and police officials killed and dozens wounded.

The deadliest attacks took place in Kirkuk, in northern Iraq. Seven people were killed and more than a dozen were wounded in 13 separate attacks that included car and roadside bombs and gunfire. In another episode, five explosives experts were killed when a bomb blew up while they were trying to defuse a rocket.

In Baghdad, a car bomb exploded amid a crowd of Shiite pilgrims, killing four people and wounding 16, while an improvised explosive device killed two Shiite pilgrims and wounded four others. In Hilla, south of the capital, improvised-explosive devices planted in civilian residences killed seven and wounded five.

A car bomb tore through the convoy of the provincial governor in Babil, killing two civilians and wounding eight, but missing the governor, Mohamed al-Messaoudi.

An improvised-explosive device also detonated inside a house north of Diyala province, killing four members of one family. In Diyala province itself, a policeman was killed and two were wounded by gunmen, while a car bomb exploded south of Samarra, killing a policeman. Three security officers were killed by gunmen in the northern city of Mosul.

Other attacks wounded 17 more people when bombers struck Shiites in Kirkuk and police officers in Tikrit. Three civilians were wounded when mortar rounds fell on houses south of Baquba.

The police have not announced possible motives for the range of attacks, and it was not clear whether there were any credible claims of responsibility. But by hitting disputed cities, politicians and both Shiites and Sunnis, some of the violence appeared to illuminate the failure to address simmering tensions among Iraq’s ethnic and sectarian groups that left a grim legacy on the final day of the first full year since U.S. forces withdrew.

In recent weeks, the troubles have only grown.

A sectarian divide between Sunnis and Shiites was on display after soldiers detained the bodyguards of the finance minister, Rafia al-Issawi, a Sunni, inciting protests last month against the Shiite-dominated government of Prime Minister Nouri al-Maliki. Leaders from a Sunni-dominated political bloc, Iraqiya, then threatened to pull out of the government and called for a no-confidence vote.

Tensions were raised to a dangerous pitch recently in Kirkuk, where soldiers squared off with Kurdish militias, after al-Maliki sought to consolidate his control over security there.