I need to correct the OP.
There are more than just 3 countries claiming the disputed islands in South China.
Besides, China (People's Republic of China), Vietnam, Philippines, there are also Brunei, Malaysia, and Taiwan (Republic of China) claiming Spartly islands.
China's claim matches Taiwan's claim exactly, since China claims all of Taiwan, and PRC's claim is based on ROC's claim.
China is indeed the first one to claim all of the Spartly islands, as well as all of the Paracel islands.
Their official map from 1947 (published by the previous regime Republic of China... still exist on Taiwan) had the 11-dash-line and later updated to 9-dash-line. These lines included almost entire South China Sea with no one objected to it at the time of publishing.
Philippines' claim came in 1950s when a Philipino Tomas Cloma went to Spartly and unrooted two flags of ROC there before he claimed the islands for himself. Philippines claimed the islands from Tomas Cloma. ROC responded angrily and immediately re-stationed troops on Spartly islands (they pulled the troops out when they lost the civil war on mainland China against the communists). The US government backed the claim of ROC at the time, and reminded Philippines that Spartly islands were not a part of the territory when Spain signed the 1898 treaty to give up Philippines to the US.
As for Vietnam, their claim was based on the claim of the French who used to occupy all of the Indochina (Vietnam was a part of it). France then recognized the ownership of these islands by China in exchange for the Qing Empire of China to recognize the "legit" ownership of Indochina by the French. They went back on their words after the fall of Qing Empire in 1911.
The bottom line is, if you look at the map, no one but China and Vietnam could potentially own Paracel islands. Parts of the Paracel islands are within EEZ of China.
As for Spartly islands, it is a little more complicated since there are countries in the vicinity.
There are two types of claim to the disputed islands. One type is based on historical facts. The other is based on some international rules passed in the last a few decades. Only China and Vietnam had based their claim on their historical facts. The others like Philippines, knew it never had the ownership of Spartly islands, but claim it anyway since the existence of EEZ came into effect in 1980s, although their first claim was based on the "private" claim of a single Philipino named Tomas Cloma who landed on Spartly in 1950s and unrooted two flags of ROC there.
Here is a quote
Japanese scholar Taoka Shunji criticized Japanese Prime Minister Shinzo Abe for trying to falsely portray China as a threat to Japan and that it was invading its neighbors like the Philippines, and pointed out that the Spratly islands were not part of the Philippines when the US acquired the Philippines from Spain in the Treaty of Paris in 1898, and the Japanese ruled Taiwan itself had annexed the Spratly islands in 1938 and the US ruled Philippines did not challenge the move and never asserted that it was their territory.
You can look at some more information on wikipedia ( I know everyone can edit wikipedia, but they do have credible sources listed at the bottom of the pages, which you can look them up one by one and judge it by yourself. )
