There were no casualties in either incident.
In the most recent attack, a grenade thrown from a passing car hit a wall outside the Indian mission, exploding on the road outside the consulate.
According to a spokesman for Afghanistan's Interior Ministry, Yousuf Stanezai, police have launched a manhunt but no one has so far been arrested.
Kandahar is a former stronghold of the Taliban, who ruled Afghanistan from 1996 until being overthrown by a U.S.-led coalition in 2002.
That attack came just hours after a grenade exploded outside the Ministry of Women's Affairs in the capital, Kabul.
The ministry had closed for the night at the time of the attack.
Taliban-led violence has increased across Afghanistan in the past year, raising fears that the insurgency may hamper moves toward democracy.