I was having the same problem just now. I solved it by uninstalling the latest Connector/NET drivers (6.7.4) and then installed the older drivers (6.6.5) and it works.
I am using Visual Studio 2010. I uninstalled the latest ones because I figured they were somehow related to .NET4.5, which I'm not able to use.
Update #1:
Supposedly another way is to register the MySql Connector with various Visual Studio versions (2010/2012/2013/2015...) during installation: Go to Modify Product Features and select all the relevant Visual Studio versions.
Update #2 - Visual Studio 2019 Update:
When I installed MySQL Community with the ConnectorNET and VisualStudio Plugin options included - MySQL didn't show up as a data provider in Visual Studio.
The installer I used included the VS Plugin version 1.2.9, which had supposedly fixed installation issues from 1.2.8, but still didn't work for me...
The solution for me was to uninstall the Connector and the Visual Studio Plugin, download them as individual components, and then install them separately (not as part of the MySQLServer Installer). Install the Connector first, then VS plugin after.
I found the solution here, Thanks to @LambertHeenan.
NOTE about Visual Studio Express
The OP asks whether MySQL is supported with Visual Studio Express (which as far as I can tell has been renamed to Visual Studio Community). In the past MySQL officially didn't support Visual Studio Express, as per @Paul's answer below, but they do officially support Visual Studio Community 2017 and 2019, according to this page.
与恶龙缠斗过久,自身亦成为恶龙；凝视深渊过久,深渊将回以凝视…
