Access SQL Server on Azure VM via SQL Server Management Studio on my local Machine

You need to complete all of the following to achieve what you want:

• Add a TCP endpoint for that VM on port 1433 - public & private (you can change the public port if you will).
• Configure the SQL Server Instance to listen to TCP.
• Enable the incomping TCP port 1433 in the local FireWall rules of the VM.
• Enable mixed mode authentication on the SQL Server
• Make sure your ISP does not block outgoing port 1433 (a common practice for most of ISVs since 2003' SQL Slammer) - this is overridable with custom Public port for your TCP endpoint.

Than you will be able to connect to the SQL Server in the Azure VM using your local SSMS.

Faced this issue with Azure VM, I can't connect to SQL server.Completed all steps described in answer above but didn't get success..

It started to work when I set specific port for TCP/IP protocol.

So open SQL Server Configuration Manager, find TCP/IP protocol for your instance, open Properties and set port to 1433 in IPAll section.

I had the same problem, what I should change was: click on the VM in AzurePortal --> SQL Server configuration --> Change SQL Connectivity to public

then it works for me!