RNZ Pacific correspondent Georgina Kekea said the government curfew ended this morning at 7am local time, and people are now milling around.
Some people are continuing to loot the shops.
Kekea said the rioters have stolen a lot of alcohol.
RNZ Pacific correspondent Elizabeth Osifelo said the destruction is focused on the city's east.
"Chinatown has now been badly affected and also the industrial area in Honiara, that has also been affected. A lot of local businesses have also gone up in flames," she said.
In a public address last night through the Solomon Islands Broadcasting Corporation, Prime Minister Manasseh Sogavare said he had been asked to step down, but this was something he will not do.
"If I have to step down, what message would that send to our people, children and generations to come?
"Some of us are of the opinion that if I step down, the protests and riots will stop, this is the easiest decision to make. However, the effect of this decision is what weighs heavy in my heart," he said.
Dozens of Australian police and defence force (AFP) personnel have begun arriving in the Solomon Islands capital amid major unrest.
Australian Prime Minister Scott Morrison said the deployment is in response to a request from Sogavare as part of a security treaty signed with Australia in 2017.
He said the 23 AFP personnel will help with riot control, and up to another 50 will be deployed to support security at critical infrastructure.
Morrison said he has spoken to the leaders of New Zealand, Fiji and Papua New Guinea to discuss how peace and stability can be restored.
"That our purpose here is to provide stability and security to enable the normal constitution processes within the Solomon Islands to be able deal with the various issues that have arisen, and that it be done in a climate of peace and security," he said.
Morrison said that Australia's assistance would not result in it interfering in the internal affairs of Solomon Islands.