I'm curious if soldiers just all immediately stopped fighting or if some kept going at it. How does a worldwide war end?
Here's one curious case of post-surrender conflict:
Steuart Lansing Pittman was born in Albany on June 6, 1919.... He was sent to China to train and operate with guerrilla groups behind Japanese lines.
Two days after V-J Day, Mr. Pittman was involved in one of the most unusual naval battles of the war, and possibly the last. Mr. Pittman was commanding two Chinese junks carrying guerrillas when they were fired on by a Japanese junk in the South China Sea. Mr. Pittman’s forces counterattacked, killing 43 and taking 39 Japanese sailors prisoner. He was awarded the Silver Star for valor.
http://www.nytimes.com/2013/02/19/us/steuart-pittman-head-of-fallout-shelter-program-dies-at-93.html
There were a lot of holdouts among the Japanese soldiers. Wikipedia has a list of later surrenders and the like, but the most famous as far as I know was in 1974, when Hiro Onoda was relieved by his former commanding officer.
He had been fighting a guerrilla war in the Philippines with a few other soldiers, and was the last one alive when he was finally captured. He didn't accept the japanese surrender until his commanding officer traveled from Japan to relieve him.
I wish I knew more about the European theatre, but the closest I can think of is that German soldiers tried to march west to avoid surrendering to the Soviet Army.
Edit: fixing the link
The Prague Offensive, the Battle of Poljana, the Battle of Odžak, the Uprising on Texel, and the Battle of Slivice all lasted past VE Day. Those that started after VE Day were due to German and Collaborationist troops attempting to surrender to the Western Allies but encountering Partisans or Soviets on the way.
The Soviets battled the Japanese after the ceasefire on August 15th with the Manchurian Offensive Operation, and their invasion of the Kuril Islands, with he last battle being the Battle of Shumshu. The Soviets Offensive was due to them being promised the Kuril Islands and Southern Sakhalin by the Western Allies with the ultimate goal of taking and annexing Hokkaido (which wasn't promised) however they were pressured into stopping short due to pressure from the Allies.