The simplest way to get # cat and # dog in the example is using -Pattern '^# '

Though, in this case it must be ensured that the line always starts with # followed by at least one whitespace. Lines with leading whitespaces and also strings directly adjacent after # without any whitespaces between will not match.

# cat
######## foo
### bar
#################### fish
# dog
##### test
   #     bird
      #monkey

For getting # cat, # dog, # bird and #monkey it's better to use:

Get-Content file.txt | Select-String -Pattern '^(\s*)#(?!#)'

The solution of using a negative lookahead (?!#) has already been mentioned, don't know why the answer was downvoted.

^(\s*) describes that at the start of the line any whitespace characters in zero or more occurrence before # will match.