m#/([^/]*)$#;
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Tip #9 from the Basic debugging checklist: YAPE::Regex::Explain. But, due to some limitations, I had to give the tool a little help:

The regular expression:

matches as follows:
  
NODE                     EXPLANATION
----------------------------------------------------------------------
(?-imsx:                 group, but do not capture (case-sensitive)
                         (with ^ and $ matching normally) (with . not
                         matching \n) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  /                        '/'
----------------------------------------------------------------------
  (                        group and capture to \1:
----------------------------------------------------------------------
    [^/]*                    any character except: '/' (0 or more
                             times (matching the most amount
                             possible))
----------------------------------------------------------------------
  )                        end of \1
----------------------------------------------------------------------
  $                        before an optional \n, and the end of the
                           string
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------
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Customarily, // are used as delimiters, but since it wants to match /, alternate delimiters are used (##) to avoid excessive escaping.

See also:

• Read Writeup Formatting Tips and update your post to add "code" tags around your code.
• perlop
• File::Basename

Sometimes re debug can help:

#!/usr/bin/perl

use strict;
use warnings;
use re 'debug';

m#/([^/]*)$#;
     
neil@neptune ~/src/perltest $ ./redebug.pl 
Compiling REx "/([^/]*)$"
Final program:
   1: EXACT </> (3)
   3: OPEN1 (5)
   5:   STAR (17)
   6:     ANYOF[^/][{above_bitmap_all}] (0)
  17: CLOSE1 (19)
  19: SEOL (20)
  20: END (0)
anchored "/" at 0 floating ""$ at 1..9223372036854775807 (checking anc
+hored) minlen 1 
Use of uninitialized value $_ in pattern match (m//) at ./redebug.pl l
+ine 7.
Freeing REx: "/([^/]*)$"
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The [^/] means anything not a /. The * means 0 or more of those characters. The $ means at the end of the line. So the whole thing will match any characters at the end of the line that are not /'s. Typically if you have a filename with its path, this will capture everything after the last slash: the filename.