How do I run two separate instances of Spyder
Update for Spyder 5.0.3+ (07/2021 onwards)
First, same as Carlos's answer:
Tools > Preferences
Application > Advanced settings
[✔] Use a single instance, so it becomes
 Use a single instance, which will result in a new window opening each time Spyder is launched.
Still working for me in Spyder 5.1.5.
You can use the --new-instance option as well. Just run
(Spyder maintainer here) This is easy. You need to go to:
Tools > Preferences > Application
in Spyder 5, or
Tools > Preferences > General
in Spyder 4, click the "Advanced Settings" tab, and deactivate the option called
[ ] Use a single instance
Then every time you start Spyder a new window will be opened. If you want the old behavior back, just activate that option again.