Following up on choroba's suggestion re. eval, you could build a hash of sort criteria whereby the sort string can be validated so as to avoid running arbitrary code.

use strict;
use warnings;

use 5.014;

my %sortCriteria = (
   field1 => {
      q{} => q{$a->[ 1 ] cmp $b->[ 1 ]},
      a   => q{$a->[ 1 ] cmp $b->[ 1 ]},
      d   => q{$b->[ 1 ] cmp $a->[ 1 ]},
      n   => q{$a->[ 1 ] <=> $b->[ 1 ]},
      na  => q{$a->[ 1 ] <=> $b->[ 1 ]},
      nd  => q{$b->[ 1 ] <=> $a->[ 1 ]},
      an  => q{$a->[ 1 ] <=> $b->[ 1 ]},
      dn  => q{$b->[ 1 ] <=> $a->[ 1 ]},
      },
   field2 => {
      q{} => q{$a->[ 2 ] cmp $b->[ 2 ]},
      a   => q{$a->[ 2 ] cmp $b->[ 2 ]},
      d   => q{$b->[ 2 ] cmp $a->[ 2 ]},
      },
   field3 => {
      q{} => q{$a->[ 3 ] cmp $b->[ 3 ]},
      a   => q{$a->[ 3 ] cmp $b->[ 3 ]},
      d   => q{$b->[ 3 ] cmp $a->[ 3 ]},
      n   => q{$a->[ 3 ] <=> $b->[ 3 ]},
      na  => q{$a->[ 3 ] <=> $b->[ 3 ]},
      nd  => q{$b->[ 3 ] <=> $a->[ 3 ]},
      an  => q{$a->[ 3 ] <=> $b->[ 3 ]},
      dn  => q{$b->[ 3 ] <=> $a->[ 3 ]},
      },
   field4 => {
      q{} => q{$a->[ 4 ] cmp $b->[ 4 ]},
      a   => q{$a->[ 4 ] cmp $b->[ 4 ]},
      d   => q{$b->[ 4 ] cmp $a->[ 4 ]},
      },
   );

my @array =
   map { map { [ split m{,} ] } split m{\n} } <<EOD;
Line1,123,cat,79,Tiddles
Line2,427,dog,298,Jack
Line3,278,aardvark,9,Digger
Line4,89,gerbil,88,Jerry
Line5,427,ferret,706,Faustus
Line6,123,goldfish,23,Fred
EOD

say q{Original array};
say q{-} x 35;
say join q{,}, @$_ for @array;
say q{=} x 35;

foreach my $sortBy ( do {
   no warnings qw{ qw };
   qw{
      field1
      field1(n)
      field1(d)
      field1(nd),field2(d)
      field2(d),field3
      field4(d)
      field3
      field3(n)
      };
   } )
{
    say qq{Sort String - $sortBy};
    say q{-} x 35;
    my $sortCode = buildSorter( $sortBy );
    my @sorted = sort $sortCode @array;
    say join q{,}, @$_ for @sorted;
    say q{=} x 35;
}


sub buildSorter
{
    my $sortStr  = shift;
    
    my @criteria =
       map { [ m{(?x) ^ ( [^(]+ ) (?: \( ([and]+) \) )? $} ] }
       split m{\s*,\s*}, $sortStr;

    foreach my $criterion ( @criteria )
    {
        die qq{Invalid field "$criterion->[ 0 ]"\n}
            unless exists $sortCriteria{ $criterion->[ 0 ] };
        $criterion->[ 1 ] = q{} unless defined $criterion->[ 1 ];
        die
          qq{Invalid modifier "$criterion->[ 1 ]"},
          qq{ for field "$criterion->[ 0 ]"\n}
          unless
             exists $sortCriteria{ $criterion->[ 0 ] }->{ $criterion->
+[ 1 ] };
    }
    return eval
         q|sub { |
       . join( q{ || },
            map { $sortCriteria{ $_->[ 0 ] }->{ $_->[ 1 ] } }
            @criteria )
       . q| }|;
}
[download]

The output

Original array
-----------------------------------
Line1,123,cat,79,Tiddles
Line2,427,dog,298,Jack
Line3,278,aardvark,9,Digger
Line4,89,gerbil,88,Jerry
Line5,427,ferret,706,Faustus
Line6,123,goldfish,23,Fred
===================================
Sort String - field1
-----------------------------------
Line1,123,cat,79,Tiddles
Line6,123,goldfish,23,Fred
Line3,278,aardvark,9,Digger
Line2,427,dog,298,Jack
Line5,427,ferret,706,Faustus
Line4,89,gerbil,88,Jerry
===================================
Sort String - field1(n)
-----------------------------------
Line4,89,gerbil,88,Jerry
Line1,123,cat,79,Tiddles
Line6,123,goldfish,23,Fred
Line3,278,aardvark,9,Digger
Line2,427,dog,298,Jack
Line5,427,ferret,706,Faustus
===================================
Sort String - field1(d)
-----------------------------------
Line4,89,gerbil,88,Jerry
Line2,427,dog,298,Jack
Line5,427,ferret,706,Faustus
Line3,278,aardvark,9,Digger
Line1,123,cat,79,Tiddles
Line6,123,goldfish,23,Fred
===================================
Sort String - field1(nd),field2(d)
-----------------------------------
Line5,427,ferret,706,Faustus
Line2,427,dog,298,Jack
Line3,278,aardvark,9,Digger
Line6,123,goldfish,23,Fred
Line1,123,cat,79,Tiddles
Line4,89,gerbil,88,Jerry
===================================
Sort String - field2(d),field3
-----------------------------------
Line6,123,goldfish,23,Fred
Line4,89,gerbil,88,Jerry
Line5,427,ferret,706,Faustus
Line2,427,dog,298,Jack
Line1,123,cat,79,Tiddles
Line3,278,aardvark,9,Digger
===================================
Sort String - field4(d)
-----------------------------------
Line1,123,cat,79,Tiddles
Line4,89,gerbil,88,Jerry
Line2,427,dog,298,Jack
Line6,123,goldfish,23,Fred
Line5,427,ferret,706,Faustus
Line3,278,aardvark,9,Digger
===================================
Sort String - field3
-----------------------------------
Line6,123,goldfish,23,Fred
Line2,427,dog,298,Jack
Line5,427,ferret,706,Faustus
Line1,123,cat,79,Tiddles
Line4,89,gerbil,88,Jerry
Line3,278,aardvark,9,Digger
===================================
Sort String - field3(n)
-----------------------------------
Line3,278,aardvark,9,Digger
Line6,123,goldfish,23,Fred
Line1,123,cat,79,Tiddles
Line4,89,gerbil,88,Jerry
Line2,427,dog,298,Jack
Line5,427,ferret,706,Faustus
===================================
[download]

I hope this is helpful.

Update: Added headers for each sort operation to make it clearer what is going on.