PHP, 98 Bytes

for(;($t=$argv[1])[++$i];)$s+=$r[]=$t[$i-1]<=>$t[$i];echo(max($r)-min($r)<2)*(0<=>$s)*(1<abs($s)); 

or

$s=str_split($t=$argv[1]);sort($s);echo(-($t==strrev($j=join($s)))|$t==$j)*!!count_chars($t,3)[2]; 

0 not wavy 1 raising -1 decreasing

PHP, 96 Bytes

for(;($t=$argv[1])[++$i];)$s+=$r[]=$t[$i-1]<=>$t[$i];echo(max($r)-min($r)<2)*(0<=>$s)*(1<$s*$s); 

or 98 Bytes

$s=str_split($t=$argv[1]);sort($s);echo(-($t==strrev($j=join($s)))|$t==$j)*!!count_chars($t,3)[2]; 

0 not wavy 1 raising -1 decreasing

PHP, 98 Bytes

$s=str_split($t=$argv[1]);sort($s);echo(-($t==strrev($j=join($s)))|$t==$j)*!!count_chars($t,3)[2]; 

0 not wavy 1 raising -1 decreasing

PHP, 98 Bytes

for(;($t=$argv[1])[++$i];)$s+=$r[]=$t[$i-1]<=>$t[$i];echo(max($r)-min($r)<2)*(0<=>$s)*(1<abs($s)); 

or

$s=str_split($t=$argv[1]);sort($s);echo(-($t==strrev($j=join($s)))|$t==$j)*!!count_chars($t,3)[2]; 

0 not wavy 1 raising -1 decreasing

PHP, 101 Bytes

$s=str_split($t=$argv[1]);sort($s);echo(2*($t==strrev($j=join($s)))|$t==$j)*(!!count_chars($t,3)[2]); 

0 not wavy 1 raising 2 decreasing

PHP, 98 Bytes

$s=str_split($t=$argv[1]);sort($s);echo(-($t==strrev($j=join($s)))|$t==$j)*!!count_chars($t,3)[2]; 

0 not wavy 1 raising -1 decreasing