In the village of Mesker-Yurt, north of Shali, a young person carrying a rucksack blew himself up at a police post, investigators said, adding that "officers and civilians were not harmed by the blast".
Police in the North Caucasus region said in another incident two men armed with knives attempted to enter the district police department in the town of Shali and "inflicted wounds on two police employees on duty" before being shot dead.
Police have shot dead four teenagers who launched three attacks in and around the Chechen capital Grozny, say officials in the Russian republic.
IS claimed responsibly for the attacks, the SITE monitoring group reported, citing the jihadis' propaganda agency, Amaq.
Dzhambulat Umarov, information minister in the regional government, told the Tass news agency the attackers were aged between 11 and 16. The attack was claimed by an Islamist group. He said ISIL has increasingly focused on teenagers in its efforts to recruit supporters. All of the attackers were part of a single group from Shali, said Karimov.
"The criminals were neutralised", the investigative committee said.
The assailant had attempted a suicide bombing but failed, Kadyrov said. Two officers were injured and the attackers were shot.
Mr Kadyrov also said that the situation in Chechen republic is "calm and stabile". State media said one of the officers later died of his injuries, but police denied the report.
In a post on Telegram the Chechen leader, Ramzan Kadyrov, condemned the teenagers who, he alleged, had tried to disrupt Muslim Chechens' celebrations of Eid al-Adha, which begins on Tuesday.
News agencies cited unnamed sources saying an assailant ran over three traffic policemen in his auto.
Chechnya's interior minister Ruslan Alkhanov said the assailants "attempted to destabilize the situation in Chechnya" but have been stopped. In another clash, a man was shot after he struck a traffic officer with his auto.
Moscow battled separatists in two brutal wars in the 1990s in Chechnya and has since installed Kadyrov, a former separatist turned Kremlin loyalist, to rule the republic with an iron fist.
Global human rights groups have accused Kadyrov of rampant abuses, including arbitrary arrests and extrajudicial killings by his feared security forces.