Command or query to pull the membership of node in multiple pools

Here's some code that may help for PowerShell iControl... Just replace the server address with the right ip/hostname and update the nodeToFind variable. It's a little wordy, but I think it'll accomplish what you're looking for. (Could be wrapped as a function as well to make it easier)

 Connect to the BIG-IP
$creds  = Get-Credential;
$server = "0.0.0.0";
$iConn  = Initialize-F5.iControl -HostName $server -Username $creds.Username -Password $creds.GetNetworkCredential().Password;
$iCtrl  = Get-F5.iControl;

 Get Node Information
$nodeNames   = @($ictrl.LocalLBNodeAddressV2.get_list());
$nodeAddress = @($ictrl.LocalLBNodeAddressV2.get_address($nodeNames));
$nodeList    = @();
for ($i=0; $i -le $nodeNames.Length; $i++) {
    $nodeList += "" | Select-Object @{Name="Name";   Expression={$nodeNames[$i]}}, @{Name="Address";Expression={$nodeAddress[$i]}}
}
 Get Pool Information
$poolNames   = @($ictrl.LocalLBPool.get_list());
$poolMembers = @($ictrl.LocalLBPool.get_member_v2($poolNames));
$poolList    = @();
for ($i=0; $i -le $poolNames.Length; $i++) {
    $poolList += "" | Select-Object @{Name="Name";   Expression={$poolNames[$i]}}, @{Name="Members";Expression={@($poolMembers[$i])}}
}

 Node to find
$nodeToFind = "10.0.1.1";  or "google.com"
$node       = $nodeList | ?{ $_.Name -like "*$nodeToFind" -or $_.Address -like $nodeToFind };

 Get associated pools
$poolList | ?{ $_.Members | ?{ $_.address -eq $node.Name } }