The two sides clashed in several southern and central areas of the capital, and opened a new frontline to the east of the main airport.
Meanwhile, two explosions shook the city as Nato jets passed overhead.
Col Gaddafi's whereabouts are unknown, but overnight he vowed in a speech to fight until victory or martyrdom.
A pro-regime television channel, al-Uruba, broadcast an audio statement in which he said his decision to leave his Bab al-Aziziya compound was a "tactical" move.
He urged Libyans to cleanse the streets of "traitors, infidels and rats", and said he had "been out a bit in Tripoli discreetly, without being seen".
All of you, sweep into Tripoli and flush it out and exterminate the traitors, infidels and rats"
End Quote
"It did not feel like Tripoli had fallen or someone had marched into it."
The rebels later offered a $1.67m (£1m) reward to anyone who captured or killed Col Gaddafi. Commanders said it was vital to do so to eliminate any chance that he and his supporters could strike back.
They insisted it was only a matter of time before he was found, but conceded they had no indication of where he might have taken refuge.
One rebel official, who gave his name as Abdul Rahman, told the Reuters news agency that it was thought that Col Gaddafi was still in Tripoli, possibly in the southern al-Hadhba area, where there was fighting.
He is also believed to retain a strong following in two other cities - Sirte, his hometown 450km (280 miles) to the east of the capital, and Sabha, 650km to the south in the desert.
Analysts say Sabha has a significant military and air force base and, if Col Gaddafi and his family can reach it, it would provide them the option of easy desert escape routes into Niger or Chad.
A rebel spokesman told the BBC that negotiations were going on with local leaders in Sabha and Sirte seeking a peaceful end to the conflict.

The rebels are using artillery to try to flush out gunmen entrenched in the centre of the compound.
Many of the loyalists are members of the fugitive leader's tribe or professional soldiers, and are prepared to fight to the death, our correspondent adds.
"There are snipers above and around the perimeter of Bab al-Aziziya; there are dozens of them but we don't know where they are," rebel commander Nouri Mohammed told the AFP news agency.
There has also been gunfire in the area around the capital's Rixos Hotel, where a number of foreign journalists have been trapped.
A Reuters reporter who was in the area around midday (10:00 GMT) said automatic weapons and anti-aircraft guns were being fired.
There were also reports that that pro-Gaddafi forces had hidden on the road to the international airport, and that the fighting had spread to the areas of Abu Salim and al-Zuwara.
Witnesses say armed residents in Tripoli are continuing to man makeshift checkpoints with the help of rebel fighters.
Law and order does not appear to have broken down and celebrations have been continuing in Green Square.
The rebel administration, known as the National Transitional Council (NTC), estimates that about 400 people have been killed and thousands injured in the battle for Tripoli since Sunday.
The International Red Cross says the fighting in Tripoli has left many civilians injured. A spokesman said doctors were finding it difficult to reach hospitals in and around the capital because of continuing battles

BBC