I ran into a scenario where I needed to validate what application pool and identity a particular service application was running as. While you can do some round about poking through the UI or through IIS, it's also pretty simple to grab this information using PowerShell.
First things first, we need to grab a list of all of our service applications. We can skip this step if you already know the name of your service application.
Get-SPServiceApplication will output a list of all service applications configured on your farm. You'll notice that by default you see three columns (DisplayName, TypeName and Id). As a result of the three columns, PowerShell will truncate the contents of the columns to allow them to all display. This means that if you have long service application names, you probably won't see the full name. To get around this, we can specify the column(s) we want to see.
Get-SPServiceApplication | Select DisplayName, Id
Once you've identified the service application you want to validate (or if you already knew it), we can create a variable containing all of the properties for that particular service application. The "Identity" parameter should match the DisplayName of the service application.
$serviceApp = Get-SPServiceApplication -Identity "My Service Application"
Alternatively, you can also use the GUID to select the service application.
$serviceApp = Get-SPServiceApplication 00000000-0000-0000-0000-000000000000
Now that we have our $serviceApp variable containing all of the properties for our service application, we can view (or set) any of the available properties. The Get-Member cmdlet allows us to view a list of all objects, properties and methods that are children of the current object. In this example, it would list all of the objects, properties and methods of our selected service application.
$serviceApp | Get-Member
We can now determine the application pool and the identity of that application pool by viewing the ApplicationPool property of our service application.
Much like we did with service applications, we first need to identify the web application that we want to work with.
Get-SPWebApplication will output a list of all web applications configured on your farm. The nice thing is that web applications are a little more intuitive because we can use the URL of the web application to identify it, and that's almost certainly something you're going to know off the top of your head. Once you've identified the web application you want to validate, create a variable containing that web application.
$webApp = Get-SPWebApplication http://mywebapp.corp.tld
Just like we did with our service application, we can use the Get-Member cmdlet to list all of the objects, properties and methods of our web application.
$webApp | Get-Member
We can now determine the application pool and the identity of that application pool by viewing the ApplicationPool property of our web application. Note that Microsoft hasn't been this consistent throughout SharePoint as a whole, and there's a lot of cases where you'll find differences between property names on objects where you'd think they would be the same. Learn to love the Get-Member cmdlet, which will always show you what you've got to work with.