Could there be some mistake in the copying of the code? I tested my code and it works (both versions). Here is my complete sample code. Be sure to use the download button underneath before you copy/paste the code so you don't copy word wrapped lines. And remember that for the code with the push you also need to remove the closing '}' of the loop.

#!/usr/bin/perl

use warnings;

%threads=('a', [1,2,3],
      'b', [2,2,3],
      'e', [3,2,3],
      'd', [4,2,3],
      'c', [5,2,3]);

$tnum=0;
foreach $thread (keys %threads) {
print "\n thread for $thread, tnum=$tnum:"; #dbg
print "\n\tthe array from hash=@{$threads{$thread}}"; #dbg
# THE ABOVE LINE CORRECTLY PRINTS THE ELEMENTS OF THE ARRAY OF THE HAS

    $tsub[$tnum] = $thread;    
    $tsize[$tnum] = @{$threads{$thread}};         
print "\n\ttsize[tnum]= $tsize[$tnum]"; #dbg
   # for ($i=0; $i<$tsize[$tnum] ; $i++) 
   # { $tlist[$tnum][$i] = $threads{$thread}[$i];
push( @{$tlist[$tnum]},  @{$threads{$thread}} );
# HERE I'M TRYING TO PRINT THE ELEMENTS ONE AT A TIME
print "\n\t\@{threads{thread}}=@{$threads{$thread}}"; 
# BUT THE ABOVE LINE PRINTS ONLY A NULL 
    
}
[download]

which gives me the following output (note that as LanX found out you don't increment tnum, so there is still a bug in this code):

Name "main::tsub" used only once: possible typo at ./t7.pl line 17.
Name "main::tlist" used only once: possible typo at ./t7.pl line 22.

 thread for e, tnum=0:
        the array from hash=3 2 3
        tsize[tnum]= 3
        {threads{thread[0]}}=3 2 3

 thread for c, tnum=0:
        the array from hash=5 2 3
        tsize[tnum]= 3
        {threads{thread[0]}}=5 2 3

 thread for a, tnum=0:
        the array from hash=1 2 3
        tsize[tnum]= 3
        {threads{thread[0]}}=1 2 3

 thread for b, tnum=0:
        the array from hash=2 2 3
        tsize[tnum]= 3
        {threads{thread[0]}}=2 2 3

 thread for d, tnum=0:
        the array from hash=4 2 3
        tsize[tnum]= 3
        {threads{thread[0]}}=4 2 3
[download]