How do I enumerate IIS websites using Powershell and find the app pool for each?

Use the webadministration module:

Import-Module WebAdministration

dir IIS:\Sites # Lists all sites
dir IIS:\AppPools # Lists all app pools and applications


# List all sites, applications and appPools

dir IIS:\Sites | ForEach-Object {

    # Web site name
    $_.Name

    # Site's app pool
    $_.applicationPool

    # Any web applications on the site + their app pools
    Get-WebApplication -Site $_.Name
}

Try the following:

[Void][Reflection.Assembly]::LoadWithPartialName("Microsoft.Web.Administration")

$sm = New-Object Microsoft.Web.Administration.ServerManager

foreach($site in $sm.Sites)
{
    $root = $site.Applications | where { $_.Path -eq "/" }
    Write-Output ("Site: " + $site.Name + " | Pool: " + $root.ApplicationPoolName)
}

The script above lists every site on the server and prints the root application pool name for each site.