In one incident, a person died and 53 others were injured in the Battamaloo area of Srinagar when rebels hurled a grenade at a security patrol.
The grenade missed its target and exploded at a bus stand, a police spokesman said.
"The condition of at least half-a-dozen civilians is serious," he said, adding they were being treated in Srinagar's main hospital.
Another civilian was killed and seven others were injured on Monday in a grenade explosion in the southern town of Anantnag, 50 km south of Srinagar.
The grenade was also said to have been hurled at a security force vehicle, but it missed the target and exploded in a busy market.
Grenade attacks occur frequently in the Himalayan region, where separatists have been fighting Indian rule for the past 14 years.
Another Muslim civilian was killed and two policemen injured in two successive overnight landmine blasts in the village of Dhurian Dhole, in southern Poonch district, police said.
And rebels killed two Indian soldiers and injured another in a late night ambush in the neighbouring forests of Topi Peer.
Police said suspected rebels shot dead two Muslims including a school student in the districts of Kupwara and Doda on Monday.
Kashmir is divided between
Pakistani and Indian control
Indian troops shot dead three rebels in Phulwama and Kupwara districts on the same day.
Meanwhile, resistance fighters are believed to have taken several villagers hostage in Pulwama district after troops surrounded a house where they were hiding.
Police said security forces laid seige to a residence in the Tharin village of Pulwama district early on Monday after a tip-off that rebels were hiding there.
"There was initial exchange of fire between the two sides that stopped after sometime," a police source said.
He said security officers had urged a few villagers to enter the house to persuade the fighters to surrender and prevent the destruction of property or casualties.
At least six villagers went inside, but did not return.
Security forces have surrounded the house, but have yet to make a rescue attempt by breaking in.
More than 39,500 people have died in Indian Kashmir since the eruption of insurgency in the region in 1989, according to Indian figures.
However, separatists put the toll between 80,000 and 100,000.