Caleston Rift
The Caleston Rift is by far the easiest place to find Platinum in the "Mass Effect" universe. All five planets in the Balor Cluster and all six in the Aysur Cluster have moderate to rich amounts of Platinum. You mine these planets for Platinum by scanning them from space until you find a deposit then launch a probe to collect the minerals.
Eagle Nebula
The Amun Cluster is the best place to go in the Eagle Nebula to find Platinum. The planets Anhur, Neith, Sekhmet and Sobek in this sector all have moderate to rich Platinum deposits. Planet Rough Tide in the Relic Cluster also has rich Platinum deposits.
Crescent Nebula
The Crescent Nebula is another good place to go to mine for Platinum. The Ondestre Cluster is the best place to start -- planets Acaeria, Maisuth and Zesmeni all have moderate to rich deposits. Planet Tarith in the Lusarn Cluster also has rich Platinum deposits.
Other Sectors
Platinum is relatively rare throughout the rest of the "Mass Effect" universe. Bothros, a planet in the Hekate Cluster of the Hades Nebula, has rich deposits. In the Hourglass Nebula; planet Daratar in the Faryar Cluster is Platinum-rich as is planet Erinle in the Osun Cluster. The only other planet in the game with moderate to rich Platinum is Utha, in the Arinlarkan Cluster of the Omega Nebula.