How can I associate a file type with a powershell script?

Use the proper tools for the job:

cmd /c assoc .fob=foobarfile
cmd /c ftype foobarfile=powershell.exe -File `"C:\path\to\your.ps1`" `"%1`"

Note that both assoc and ftype are CMD-builtins, so you need to run them via cmd /c from PowerShell.

For files without extension use this association:

cmd /c assoc .=foobarfile

I don't think you can do it through Windows UI.

The goal here is to associate a type with powershell.exe, arguments to which will be

1. The powershell script
2. The target filename

To do this

1. Launch Regedit.exe. //disclaimer: you are editing Windows registry. Here be tigers.
2. Go to HKEY_CLASSES_ROOT (admin access, for all users) or HKEY_CURRENT_USER\SOFTWARE\Classes
3. Create a key named .<extension>, e.g. if you want to associate *.zbs - create a key .zbs
4. Set it's (default) value to something like zbsfile -- this is a reference linking your extension to a filetype.
5. Create a key called zbsfile - this is your filetype
6. Set (default) value to something readable, e.g. "This file is ZBS."
7. Underneath create a tree of keys (examples are all around):

zbsfile shell open command

8. Under command, set (default) value to e.g.
   powershell.exe -File "C:\path\to your\file.ps1" "%1"
   where %1 means the file user clicked

That should work.

EDIT: or (crazy idea), create a bat file doing just powershell.exe -File "C:\path\to your\file.ps1" "%%1" and select it in Windows UI...

For those like me who got here looking for general file types associations, I ended up using this function:

Function Create-Association($ext, $exe) {
    $name = cmd /c "assoc $ext 2>NUL"
    if ($name) { # Association already exists: override it
        $name = $name.Split('=')[1]
    } else { # Name doesn't exist: create it
        $name = "$($ext.Replace('.',''))file" # ".log.1" becomes "log1file"
        cmd /c 'assoc $ext=$name'
    }
    cmd /c "ftype $name=`"$exe`" `"%1`""
}

I struggled with proper quoting from @Ansgar Wiechers's answer but finally got it right :)