UPDATE: at the time the question was posted on 2018 the procedure to create a free database with PowerShell was the only one available. Starting September 26, 2023, Azure offers a free Azure SQL Database (in Preview) with a limit 100,000 vCore seconds of compute every month. This free offer provides a General Purpose database for the lifetime of the subscription. This offer is rolling out by region, with new regions added every week. The following regions are currently supported: East US, West Central US, UK South.
Azure SQL Database allows you to create one free Azure SQL Server database per region, it reverts to Standard after 365 days. That is what I know ever since you use PowerShell to create it (I don't know about using other methods).
New-AzureRmSqlDatabase -DatabaseName $DatabaseName -ServerName $sqlServer.ServerName
-ResourceGroupName $resourceGroup.ResourceGroupName -Edition 'Free'
-RequestedServiceObjectiveName 'Free'
This works only for the following regions, for now:
Location DisplayName
-------- -------------------
australiaeast Australia East
australiasoutheast Australia Southeast
centralindia Central India
eastasia East Asia
francecentral France Central
japaneast Japan East
japanwest Japan West
koreacentral Korea Central
koreasouth Korea South
northcentralus North Central US
northeurope North Europe
southeastasia Southeast Asia
southindia South India
uksouth UK South
ukwest UK West
westcentralus West Central US
westeurope West Europe
westindia West India
westus2 West US 2