tl;dr
PS> "31222829" -replace '[0-9]', '$& '
3 1 2 2 2 8 2 9
Note that the output string has a trailing space, given that each digit in the input ([0-9]) is replaced by itself ($&) followed by a space.
As for what you tried:
"31222829" -replace ("[0-9]","$0 ")
While enclosing the two RHS operands in (...) doesn't impede functionality, it's not really helpful to conceive of them as an array - just enumerate them with ,, don't enclose them in (...).
Generally, use '...' rather than "..." for the RHS operands (the regex to match and the replacement operand), so as to prevent confusion between PowerShell's string expansion (interpolation) and what the -replace operator ultimately sees.
Case in point: Due to use of "..." in the replacement operand, PowerShell's string interpolation would actually expand $0 as a variable up front, which in the absence of a variable expands to the empty string - that is why you ultimately saw a blank string.
Even if you had used '...', however, $0 has no special meaning in the replacement operand; instead, you must use $& to represent the matched string, as explained in this answer.
To unconditionally separate ALL characters with spaces:
Drew's helpful answer definitely works.
Here's a more PowerShell-idiomatic alternative:
PS> [char[]] '31222829' -join ' '
3 1 2 2 2 8 2 9
Casting a string to [char[]] returns its characters as an array, which -join then joins with a space as the separator.
Note: Since -join only places the specified separator (' ') between elements, the resulting string does not have a trailing space.