The suggested Scalar::Util::looks_like_number() is often a good choice; however, be aware of some gotchas:

Infinity (see "perldata: Special floating point: infinity (Inf) and not-a-number (NaN)"):

perl -E '
    use Scalar::Util "looks_like_number";

    for (qw{Inf -Inf Infinity -Infinity}) {
        say "$_: ", looks_like_number($_) ? 1 : 0;
    }
'
Inf: 1
-Inf: 1
Infinity: 1
-Infinity: 1
[download]

Non-decimal numbers:

perl -E '
    use Scalar::Util "looks_like_number";

    for (qw{1 0b10 0o10 0x10}) {
        say "$_: ", looks_like_number($_) ? 1 : 0
    }
'
1: 1
0b10: 0
0o10: 0
0x10: 0
[download]

perl -E '
    use Scalar::Util "looks_like_number";

    for (1, 0b10, 0o10, 0x10) {
        say "$_: ", looks_like_number($_) ? 1 : 0
    }
'
1: 1
2: 1
8: 1
16: 1
[download]

"I would like to combine the string comparisons into as few as possible but seem to break the code every time I try."

If looks_like_number() is causing you problems, perhaps due to "gotchas" indicated above, or as a purely academic exercise, here's some hints for a hand-crafted solution.

Get rid of the blacklist: this will take ages to get right and, even then, there's a high chance that you'll miss an edge-case or two (or more). Just use a whitelist which represents valid numbers for your application.

Use a single regex with alternations, and only define it once (not in every iteration). With Perl v5.10 or higher, you can use this format:

sub IsNumber {
    state $re = qr{...};
    return $_[0] =~ $re;
}
[download]

For Perls of an older vintage:

{
    my $re; BEGIN { $re = qr{...} }

    sub IsNumber {
        return $_[0] =~ $re;
    }
}
[download]

Don't try to write your regex in the smallest possible space. This isn't a golfing exercise. Readability and maintainability will very quickly deteriorate. Use the /x modifier — or /xx if available; requires Perl v5.26 or later — as explained in "perlre: /x and /xx".

Consider whether you're allowing exponents:

$ perl -E 'say 1e6'
1000000
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What about when underscores have been used to improve the readability of large numbers:

$ perl -E 'say 1_000_000/1_000'
1000
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Are you only dealing with 7-bit ASCII digits (/[0-9]/) or allowing the handling of all (i.e. Unicode®) digits (/\p{Digit}/)?

Note that /\d/ and /[[:digit:]]/ are not necessarily the same as /[0-9]/. See "perlre: /a (and /aa)".

More general sources of reference are: perlre; perlrebackslash; and, perlrecharclass.