/^[+-]?(?=\d|\.)\d*\.?\d*(?:[eE][+-]?\d+)?$/

Update: Fixed to avoid matching "+."

/^[+-]?(?=\.?\d)\d*\.?\d*(?:[eE][+-]?\d+)?$/

Update: My test code with additional cases.

#!/usr/bin/perl -w
use strict;

my @good = qw( 0e0 0 +0 -0 1.  0.14 .14 1.24e5 24e5 -24e-5);
my @bad = ('',
           qw(. +. 1e e1 2.3. 2.3.4 1..2 .1.1 e.1 e1.1 .1a 1.a 1.1a 1a
+.1 1.a1));

my $pat = qr/^[+-]?(?=\.?\d)\d*\.?\d*(?:[eE][+-]?\d+)?$/;

print "GOOD Tests (Should match.)\n";
print /$pat/ ? '   ': 'no ', "match: '$_'\n" for @good;

print "\nBAD Tests (Should not match.)\n";
print /$pat/ ? '   ': 'no ', "match: '$_'\n" for @bad;
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Update: This can be made even shorter than the solution posted by an Anonymous Monk if you change it to use capturing parens and the /i modifier instead of the [Ee] character class as he does in his.

/^[+-]?(?=\.?\d)\d*\.?\d*(e[+-]?\d+)?$/i

In response to and agreement with demerphq's reply, I don't actually recommend this last one at all. I just wanted to show that it could be golfed further.

-sauoq
"My two cents aren't worth a dime.";