(Adds separatists saying five killed, more on Hagel)
* Ukraine says separatists using heavy weapons, pilots killed
* Putin spokesman says "punitive operation" destroys deal
* Reuters reporters say military helicopter opened fire
* Man shot dead in Odessa disturbances
By Maria Tsvetkova
SLAVIANSK, Ukraine, May 2 (Reuters) - Pro-Russian rebels shot down two Ukrainian helicopters near separatist-held Slaviansk on Friday, killing two crew members, while Moscow accused Kiev of wrecking hopes of peace by launching a "criminal" assault to retake the eastern town.
Separatists said Ukrainian forces killed three of their fighters and two civilians when they moved in on Slaviansk in the early hours, but that the troops had only taken five checkpoints from the rebels.
On the other side of Ukraine, police said a man was shot dead in clashes between a crowd backing Kiev and pro-Russian activists in the largely Russian-speaking southern port of Odessa, which lies west of Crimea, annexed by Moscow in March.
President Vladimir Putin's spokesman said Kiev had fired on civilians from the air in a "punitive operation" in Slaviansk that destroyed an international peace plan. Moscow has tens of thousands of troops massed on the border and claims the right to invade if needed to protect Russian speakers.
The Western-backed government in Kiev said the use of missiles that brought down its helicopters was evidence that Russian forces were in the town. Moscow denies that its troops are on the ground.
Ukraine's acting president also said Russian "armed saboteurs" had tried to enter the country overnight, but were pushed back by Ukrainian border troops. He gave no further details. Russia's Security Service said his report was untrue.
Kiev said Moscow was backing groups in eastern Ukraine who were "putting civilians in danger, seizing hostages and creating an atmosphere of terror and violence".
Reuters journalists in Slaviansk, the most heavily fortified bastion of pro-Russian separatists in eastern Ukraine, heard shooting break out and saw one helicopter opening fire before dawn. Ten hours later, the city was largely quiet, with shops shut and armed separatists in control of the streets.
Advancing Ukrainian forces in armoured vehicles took up positions closer in to the suburbs, but rebels still controlled most of the town of 130,000.
Acting President Oleksander Turchinov said the operation had been complicated by the rebels' use of human shields and had not progressed as quickly as had been hoped.
SOUND OF CANNON
The growing chaos is overshadowing a presidential election the pro-Western leadership in Kiev is planning for May 25. The rebels are planning a vote on May 11 to seek a mandate to break with Kiev, like one held in Crimea before Moscow took it over.
For Russians, the Kremlin's rhetoric about what it calls the "fascists" in Kiev evokes the depredations of Nazi German invaders in World War Two, a regular theme on state media ahead of next week's anniversary of the Soviet victory.
On the square outside city hall in Slaviansk, about 100 people gathered on Friday and said they were appealing to Putin to send troops to help them.
Businesswoman Tamara Voshchanaya said: "What can you think when the sound of cannon makes you jump out of bed, when helicopters are flying over and shooting at our guys?
On the town's southern outskirts, eight Ukrainian armoured personnel carriers cut off the road but faced a cordon two deep of local residents shouting at them to go home.
Some rebels threw up new barricades of felled trees.
Putin's popularity has soared with the seizure of Crimea and talk of restoring Moscow's former empire. This week he restored the Soviet-era tradition of holding a May Day parade on Red Square, where marchers carried banners hailing the acquisition of Ukrainian territory.
U.S. Defense Secretary Chuck Hagel said Russia's actions in Ukraine had shattered the myth of European security in the post-Cold War era and said NATO allies had increased the danger by failing to meet their defence spending pledges.
The European Union said it was watching events in eastern Ukraine with growing concern. But Kiev is not a member of NATO and Western leaders have made clear they will not fight to defend Ukraine.
The Ukrainian Defence Ministry said two Mi-24 attack helicopters had been shot down by shoulder-launched anti-aircraft missiles while on patrol overnight around Slaviansk. Two airmen were killed and others wounded.
Other Ukrainian officials and the separatist leader in Slaviansk said earlier that one airman was taken prisoner.
A third helicopter, an Mi-8 transport aircraft, was also hit and a serviceman wounded, the Defence Ministry said. The SBU security service said this helicopter was carrying medics.
Ukrainian officials said their troops overran rebel checkpoints and Slaviansk was now "tightly encircled".
Putin's spokesman heaped blame on the Ukrainian government, which took power two months ago after pro-Western protests forced the Kremlin-backed elected president to flee to Russia.
Noting that Putin had warned before that any "punitive operation" would be a "criminal act", Dmitry Peskov told Russian news agencies that this was what had now happened at Slaviansk.
Peskov said Moscow had not heard from an envoy, Vladimir Lukin, sent to southeast Ukraine to negotiate the release of European military observers held by the rebels, but RIA Novosti news agency said later his aide had confirmed he was fine.
"While Russia is making efforts to de-escalate and settle the conflict, the Kiev regime has turned to firing on civilian towns with military aircraft and has begun a punitive operation, effectively destroying the last hope of survival for the Geneva accord," he said, referring to a deal on April 17 signed by Russia, Ukraine, the United States and the European Union.
Under that agreement, separatists were supposed to lay down their arms and vacate the public buildings they have seized in about a dozen towns they have seized across the Russian-speaking east. Since then, however, they have tightened their grip.
Ukraine's Interior Ministry said it persuaded separatists to leave two buildings in the city of Luhansk on Friday.
The SBU said the deadly use by the separatists of shoulder-launched anti-aircraft missiles was evidence that "trained, highly qualified foreign military specialists" were operating in the area "and not local civilians, as the Russian government says, armed only with guns taken from hunting stores".
On his Facebook page, Interior Minister Arsen Avakov posted: "The goal of our anti-terrorist operation and, at the same time, our demands to the terrorists are simple:
"Free the hostages, lay down weapons, vacate administrative buildings and get municipal infrastructure back to normal."
The rebels said they had the upper hand.
"They wanted to carry out some small-scale tactical operations just to scare the people," said a militant manning a checkpoint leading to the army-held airfield.
"But so far things have not worked out the way they wanted." (Additional reporting by Natalia Zinets and Elizabeth Piper in Kiev and Matthew Robinson in Donetsk; writing by Alastair Macdonald and Philippa Fletcher; editing by Peter Graff and Giles Elgood)