In PowerShell I have tried:
alias | select-string Alias
This fails even though Alias is clearly in the output. I know this is because select-string is operating on some object and not the actual output string. 
What can be done about it?
In PowerShell I have tried:
alias | select-string Alias
This fails even though Alias is clearly in the output. I know this is because select-string is operating on some object and not the actual output string. 
What can be done about it?
I think this solution is easier and better, use directly the function findstr:
alias | findstr -i Write
You can also make an alias to use grep word:
new-alias grep findstr
Your problem is that alias emits a stream of AliasInfo objects, rather than a stream of strings. This does what I think you want.
alias | out-string -stream | select-string Alias
or as a function
function grep {
  $input | out-string -stream | select-string $args
}
alias | grep Alias
When you don't handle things that are in the pipeline (like when you just ran 'alias'), the shell knows to use the ToString() method on each object (or use the output formats specified in the ETS info).
If you truly want to "grep" the formatted output (display strings) then go with Mike's approach. There are definitely times where this comes in handy. However if you want to try embracing PowerShell's object pipeline nature, then try this. First, check out the properties on the objects flowing down the pipeline:
PS> alias | Get-Member
   TypeName: System.Management.Automation.AliasInfo
Name                MemberType     Definition
----                ----------     ----------
Equals              Method         bool Equals(System.Object obj)
GetHashCode         Method         int GetHashCode()
GetType             Method         type GetType()
ToString            Method         string ToString()
<snip>
*Definition*        Property       System.String Definition {get;}
<snip>
Note the Definition property which is a header you see when you display the output of Get-Alias (alias) e.g.:
PS> alias
CommandType     Name           *Definition*
-----------     ----           ----------
Alias           %              ForEach-Object
<snip>
Usually the header title matches the property name but not always. That is where using Get-Member comes in handy. It shows you what you need to "script" against. Now if what you want to "grep" is the Definition property contents then consider this. Rather than just grepping that one property's value, you can instead filter each AliasInfo object in the pipepline by the contents of this property and you can use a regex to do it e.g.:
PS> alias | Where-Object {$_.Definition -match 'alias'}
CommandType     Name                   Definition
-----------     ----                   ----------
Alias           epal                   Export-Alias
Alias           gal                    Get-Alias
Alias           ipal                   Import-Alias
Alias           nal                    New-Alias
Alias           sal                    Set-Alias
In this example I use the Where-Object cmdlet to filter objects based on some arbitrary script. In this case, I filter by the Defintion property matched against the regex 'alias'. Only those objects that return true for that filter are allowed to propagate down the pipeline and get formatted for display on the host.
BTW if you're typing this, then you can use one of two aliases for Where-Object - 'Where' or '?'. For example:
PS> gal | ?{$_.Definition -match '-Item*'}
There are two problems. As in the question, select-string needs to operate on the output string, which can be had from "out-string". Also, select-string doesn't operate linewise on strings that are piped to it. Here is a generic solution
(alias|out-string) -split "`n" | select-string Write 
The proposed solution is just to much work for something that can be done like this:
Get-Alias -Definition Write*
Try this:
PS C:\> ipconfig /displaydns | Select-String -Pattern 'www.yahoo.com' -Context 0,7
>     www.yahoo.com
      ----------------------------------------
>     Record Name . . . . . : www.yahoo.com
      Record Type . . . . . : 5
      Time To Live  . . . . : 27
      Data Length . . . . . : 8
      Section . . . . . . . : Answer
      CNAME Record  . . . . : new-fp-shed.wg1.b.yahoo.com
PS> alias | Where-Object {$_.Definition -match 'alias'}
CommandType     Name                   Definition
-----------     ----                   ----------
Alias           epal                   Export-Alias
Alias           gal                    Get-Alias
Alias           ipal                   Import-Alias
Alias           nal                    New-Alias
Alias           sal                    Set-Alias
what would be contradict of match in PS then as in to get field not matching a certain value in a column
For a more flexible and lazy solution, you could match all properties of the objects. Most of the time, this should get you the behavior you want, and you can always be more specific when it doesn't. Here's a grep function that works based on this principle:
Function Select-ObjectPropertyValues {
    param(
    [Parameter(Mandatory=$true,Position=0)]
    [String]
    $Pattern,
    [Parameter(ValueFromPipeline)]
    $input)
    $input | Where-Object {($_.PSObject.Properties | Where-Object {$_.Value -match $Pattern} | Measure-Object).count -gt 0} | Write-Output
}