I've been fighting with a SQl Server Express installation for Personal SDE. Man, it's been painful.
At any rate, I've finally got the instance up and running on a virtual machine; when I log on to that machine and fire up Server Manager, I see the instance just fine and I can create a SQl database in it. The problem is I cannot access the instance from my desktop through ArcCatalog, which is where I have an ArcGIS Advanced license so I can't create a Geodatabase. I can't access it from a local install of Sql Manager either.
The errors I see are pretty generic, and searching google on them, it appears that while they are common, the solutions are relatively simple; IP properties to allow remote connections, etc. I entered my domain\username as a login and gave myself dbadmin privelges on the SQl side of things, using windows authentication. I must still be missing something because it's still a no-go.
This is SQL Server Express 2008 with ArcGIS 10.2.2. Any insight is appreciated.