... the format somewhat changes with each response ... no established list of all the possibilities. ... they will through a wrench in the works ...

Kinda hard to do pattern matching when there's no pattern. :) Maybe something like the following. Also, please see How to ask better questions using Test::More and sample data and Short, Self-Contained, Correct Example for useful ways of asking questions — both of the monks and of yourself! (Update: Also also, see haukex's Building Regex Alternations Dynamically article for a discussion of building the  $rx_modifier regex.)

c:\@Work\Perl\monks>perl
use strict;
use warnings;

use Test::More 'no_plan';
use Test::NoWarnings;

my @Tests = (
  [ 'https://api.weather.gov/icons/land/day/tsra_sct,20/tsra_sct,40?si
+ze=medium', 'tsra', 'tsra', ],
  [ 'https://api.weather.gov/icons/land/day/rain_showers,30/tsra_hi,30
+?size=medium', 'rain', 'tsra', ],
  [ 'https://api.weather.gov/icons/land/night/rain_showers,30/rain_sho
+wers?size=medium', 'rain', 'rain', ],
  [ 'https://api.weather.gov/icons/land/day/bkn?size=medium', 'bkn', ]
+,
  [ 'https://api.weather.gov/icons/land/day/size=medium', ],
  );

my $rx_pre  = qr{ (?<= /)   }xms;
my $rx_post = qr{ (?= [_?]) }xms;

my $rx_generic_modifier = qr{ [[:lower:]]+ }xms;

my @modifiers = qw(tsra rain sleet bkn skc few ovc);

my ($rx_modifier) =
    map  qr{ $rx_pre (?: $_ | $rx_generic_modifier) $rx_post }xms,
    join '|',
    map  quotemeta,
    reverse sort
    @modifiers
    ;
# print "modifiers regex: $rx_modifier \n";  # for debug

VECTOR:
for my $ar_vector (@Tests) {

    if (not ref $ar_vector) {
        note $ar_vector;
        next VECTOR;
        }

    my ($string, @expected_modifiers) = @$ar_vector;

    my @got_modifiers = $string =~ m{ $rx_modifier }xmsg;
    is_deeply \@got_modifiers, \@expected_modifiers,
        "'$string' -> (@got_modifiers)"
        ;

    }  # end for VECTOR

done_testing;

__END__
ok 1 - 'https://api.weather.gov/icons/land/day/tsra_sct,20/tsra_sct,40
+?size=medium' -> (tsra tsra)
ok 2 - 'https://api.weather.gov/icons/land/day/rain_showers,30/tsra_hi
+,30?size=medium' -> (rain tsra)
ok 3 - 'https://api.weather.gov/icons/land/night/rain_showers,30/rain_
+showers?size=medium' -> (rain rain)
ok 4 - 'https://api.weather.gov/icons/land/day/bkn?size=medium' -> (bk
+n)
ok 5 - 'https://api.weather.gov/icons/land/day/size=medium' -> ()
1..5
ok 6 - no warnings
1..6
[download]

Update: Moved link to haukex's article into lexical contiguity with other documentation links.