Ukraine says it has recaptured ground in Bakhmut, a rare advance after months of grinding Russian gains in the eastern city.
Kyiv said its forces advanced 2km (1.2 miles) in a week. Russia said its troops had regrouped in one area.
The claims signal a momentum shift in Bakhmut - but more widely, there is no clear evidence of an Ukrainian counter-offensive.
However, two explosions were reported on Friday in Russian-occupied Luhansk.
Images posted on social media, verified by the BBC, show a big plume of black smoke rising from the city, which lies about 90km (55.9 miles) behind the front line in eastern Ukraine.
The blasts come a day after the UK said it had supplied Ukraine with long range Storm Shadow cruise missiles.
Kremlin-appointed officials said six children in Luhansk were injured in a missile attack alongside Russian parliamentarian Viktor Vodolatsky. The authorities there have blamed the attack on Kyiv.
Luhansk is beyond the reach of the Himars rockets Ukraine has previously relied on for deep strikes against Russian targets.
But Russian-appointed officials in the region said they thought Ukrainian-made missiles were responsible, hitting administrative buildings of two defunct enterprises.
Earlier Russia's defence ministry said Russian troops in one Bakhmut area had changed their position for strategic reasons.
It said units of the southern group of Russian forces had taken up a better defensive position in the Maloilinivka area, something which took into consideration "the favourable conditions of the Berkhivka reservoir".
However the head of Russia's Wagner mercenary group Yevgeny Prigozhin said what the Ministry of Defence was talking about "is unfortunately called 'fleeing' and not a 'regrouping'".
As the intense, bloody battle has worn on, Bakhmut has become symbolically important - though many experts question its tactical value.
In a post on Telegram, Deputy Defence Minister Hanna Malyar claimed Russia suffered significant troop losses as Ukraine gained 2km without losing any positions.
Meanwhile Russian military bloggers reported Ukrainian advances or troop movements in several areas.
The Institute for the Study of War also said Ukrainian forces had probably made gains of 2km in Bakhmut.
The BBC has verified video of soldiers with Ukrainian-identifying markings posing in front of a gate and a tank in the distance, also with Ukrainian markings.
The video, published on 11 May, has been located to an area around Bakhmut industrial college, until recently held by Wagner troops.
Ukraine's President Volodymyr Zelensky says his army needs more equipment ahead of counter-offensive
Away from Bakhmut, the exiled mayor of Melitopol reported a large explosion on Friday morning in the centre of the south-eastern city, which has been occupied by Russia since the start of the war.
It was not known what caused the blast, but the Ukrainian air force made 14 strikes on Russian forces and military equipment on Thursday, Ukraine's armed forces said.
Alongside the air strikes, Ukraine said it destroyed nine Russian drones and carried out successful attacks on dozens of military targets - including artillery units, an ammunition warehouse and air defence equipment.
After months of stalemate, a Ukrainian counter-offensive - helped by newly-arrived Western weapons - has been openly discussed. But Ukraine's president said on Thursday it was too early to start the attack.
"With [what we already have] we can go forward and, I think, be successful," President Volodymyr Zelensky said in an interview.
"But we'd lose a lot of people. I think that's unacceptable. So we need to wait. We still need a bit more time."
Despite President Zelensky's words, pro-Kremlin Russian war correspondent Sasha Kots claimed the counter-offensive had begun.
Ukrainian tanks were on the Kharkiv ring road heading towards the border with Russia, he said, quoting "trusted" sources. His claims could not be independently verified.
"There are low loaders in the columns carrying Western [tank] models among others," Kots added.
"In other words," he said, "Kiev [Kyiv] has decided to aggravate the situation along the northern front in parallel with the start of offensive actions on the flanks of Artyomovsk [the Russian name for Bakhmut]."
Another Russian war correspondent, Alexander Simonov, wrote on Telegram that Ukrainian forces had broken through near the village of Bohdanivka, close to Bakhmut, taking "several square kilometres" of ground.
Ukrainian military analyst Oleksandr Musivenko said Kyiv recognised that the anticipated counter-offensive might not necessarily defeat Russia "in all occupied areas".
He told Ukrainian NV radio there was every possibility the war could continue into next year. "It all depends on how the battles develop. We can't guarantee how the counteroffensive will develop," he said.
An unnamed senior US military official told CNN that Ukrainian forces were preparing for a major counter-offensive by striking targets such as weapons depots, command centres and armour and artillery systems.
Ukraine's spring 2022 advances in the southern and north-eastern parts of the country were also preceded by air attacks to "shape" the battlefield.
BDST: 1436 HRS, MAY 13, 2023