0.3mg is the physiological dose: in dose-ranging polysomnography work it restored sleep efficiency and returned plasma melatonin to the normal nocturnal range, while 3.0mg induced hypothermia and left plasma melatonin elevated into daylight hours (PMID: 11600532).
Doses studied across this literature run from 0.1mg to 10mg.
For jet lag, 0.5 to 5mg were similarly effective and doses above 5mg were no more effective, and a 2mg slow-release form was relatively ineffective, which suggests a short-lived higher peak suits that use (PMID: 11279722).
Prolonged-release 2mg is the form tested for nightly use over 3 weeks to 12 months (PMID: 17875243, 21091391, 21845053), and Circadin 2mg prolonged-release is licensed in Europe for primary insomnia in adults aged 55 and over (PMID: 21845053).
Across the meta-analyses the direction of effect is consistent, but the average size is modest: roughly 4 to 7 minutes off sleep onset latency (PMID: 15649737, 23691095).