PowerShell 7 and later

PowerShell 7 introduces many new features and migrates from .NET Framework to .NET Core. As of mid-2020, it hasn't completely replaced legacy versions of PowerShell due to the reliance on .NET Core, but Microsoft has indicated that they intend for the Core family to eventually replace the legacy Framework family. By the time you read this, a compatible version of PowerShell may come pre-installed on your system; if not, see https://github.com/powershell/powershell.

Per the documentation, the following operators are supported out-of-the-box in PowerShell 7.0:

1. Null-coalescing: ??
2. Null-coalescing assignment: ??=
3. Ternary: ... ? ... : ...

These work as you would expect for null coalescing:

a ?? c ?? 'default value'
$y ??= 'default value'

Since a ternary operator has been introduced, the following is now possible, though it's unnecessary given the addition of a null coalescing operator:

a -eq $null ? a

As of 7.0, the following are also available if the PSNullConditionalOperators optional feature is enabled, as explained in the docs (1, 2):

1. Null-conditional member access for members: ?.
2. Null-conditional member access for arrays et al: ?[]

These have a few caveats:

1. Since these are experimental, they're subject to change. They may no longer be considered experimental by the time you read this, and the list of caveats may have changed.
2. Variables must be enclosed in ${} if followed by one of the experimental operators because question marks are permitted in variable names. It's unclear if this will be the case if/when the features graduate from experimental status (see issue #11379). For example, ${x}?.Test() uses the new operator, but $x?.Test() runs Test() on a variable named $x?.
3. There is no ?( operator as you might expect if you're coming from TypeScript. The following won't work: $x.Test?()

PowerShell 6 and earlier

PowerShell versions prior to 7 do have an actual null coalescing operator, or at least an operator that is capable of such behavior. That operator is -ne:

# Format:
# (b, null)[0]
($null, 'alpha', 1 -ne $null)[0]

# Output:
alpha

It's a bit more versatile than a null coalescing operator, since it makes an array of all non-null items:

$items = $null, 'alpha', 5, 0, '', @(), $null, $true, $false
$instances = $items -ne $null
[string]::Join(', ', ($instances | ForEach-Object -Process { $_.GetType() }))

# Result:
System.String, System.Int32, System.Int32, System.String, System.Object[],
System.Boolean, System.Boolean

-eq works similarly, which is useful for counting null entries:

($null, 'a', $null -eq $null).Length

# Result:
2

But anyway, here's a typical case to mirror C#'s ?? operator:

'Filename: {0}' -f ($filename, 'Unknown' -ne $null)[0] | Write-Output

Explanation

This explanation is based on an edit suggestion from an anonymous user. Thanks, whoever you are!

Based on the order of operations, this works in following order:

1. The , operator creates an array of values to be tested.
2. The -ne operator filters out any items from the array that match the specified value--in this case, null. The result is an array of non-null values in the same order as the array created in Step 1.
3. [0] is used to select the first element of the filtered array.

Simplifying that:

1. Create an array of possible values, in preferred order
2. Exclude all null values from the array
3. Take the first item from the resulting array

Caveats

Unlike C#'s null coalescing operator, every possible expression will be evaluated, since the first step is to create an array.