I create an empty hash table and then populate it with the results

Set a results variable to the names from the hash table and count the number of records

and call it like this

Note that the search uses the contains method so no need for wildcards

Results come out like this

#############################################################################    ################
#
# NAME: Find-Database.ps1
# AUTHOR: Rob Sewell http://newsqldbawiththebeard.wordpress.com
# DATE:22/07/2013
#
# COMMENTS: Load function for finding a database
# USAGE: Find-Database DBName
##################################


Function Find-Database ([string]$Search) {

    [System.Reflection.Assembly]::LoadWithPartialName('Microsoft.SqlServer.    SMO') | out-null

    # Pull a list of servers from a local text file

    $servers = Get-Content 'sqlservers.txt'

    #Create an empty Hash Table
    $ht = @{}
    $b = 0

    #Convert Search to Lower Case
    $DatabaseNameSearch = $search.ToLower()  

				Write-Output "#################################"
				Write-Output "Searching for $DatabaseNameSearch "  
				Write-Output "#################################"  

                                 

    #loop through each server and check database name against input
                    
    foreach ($server in $servers) {

        if (Test-Connection $Server -Count 1 -Quiet) {
            $srv = New-Object ('Microsoft.SqlServer.Management.Smo.Server')     $server
    
            foreach ($database in $srv.Databases) {
                $databaseName = $database.Name.ToLower()

                if ($databaseName.Contains($DatabaseNameSearch)) {

                    $DatabaseNameResult = $database.name
                    $Key = "$Server -- $DatabaseNameResult"
                    $ht.add($Key , $b)
                    $b = $b + 1
                }
            }        
        }
    }

    $Results = $ht.GetEnumerator() | Sort-Object Name|Select Name
    $Resultscount = $ht.Count

    if ($Resultscount -gt 0) {

        Write-Output "###############   I Found It!!  #################"
        foreach ($R in $Results) {
            Write-Output $R.Name 
        }
    }
    Else {
        Write-Output "############    I am really sorry. I cannot find"      $DatabaseNameSearch  "Anywhere  ##################### "
    }             
}