On linux /bin/sh is normally linked to /bin/bash or /bin/dash
both of which support the POSIX $(..) and $((..)) syntax.

The patch below is wrong really. The correct fix is to
default the is_posix variable to 1. Others disagree however :(
http://groups.google.com/group/vim_dev/t/41139a32772b2f5f

--- sh.vim.orig	2008-06-16 06:38:39.000000000 +0000
+++ sh.vim	2008-06-16 06:47:21.000000000 +0000
@@ -219,18 +219,11 @@
 syn region  shCommandSub   start="`" skip="\\\\\|\\." end="`" contains=@shCommandSubList
 
 " $() and $(()): {{{1
-" $(..) is not supported by sh (Bourne shell).  However, apparently
-" some systems (HP?) have as their /bin/sh a (link to) Korn shell
-" (ie. Posix compliant shell).  /bin/ksh should work for those
-" systems too, however, so the following syntax will flag $(..) as
-" an Error under /bin/sh.  By consensus of vimdev'ers!
-if exists("b:is_kornshell") || exists("b:is_bash")
- syn region shCommandSub matchgroup=shCmdSubRegion start="\$("  skip='\\\\\|\\.' end=")"  contains=@shCommandSubList
- syn region shArithmetic matchgroup=shArithRegion  start="\$((" skip='\\\\\|\\.' end="))" contains=@shCommandSubList
- syn match  shSkipInitWS contained	"^\s\+"
-else
- syn region shCommandSub matchgroup=Error start="\$(" end=")" contains=@shCommandSubList
-endif
+" $(..) is not supported by the original Bourne shell,
+" but is Posix compliant and so supported by all contemporary shells
+syn region shCommandSub matchgroup=shCmdSubRegion start="\$("  skip='\\\\\|\\.' end=")"  contains=@shCommandSubList
+syn region shArithmetic matchgroup=shArithRegion  start="\$((" skip='\\\\\|\\.' end="))" contains=@shCommandSubList
+syn match  shSkipInitWS contained	"^\s\+"
 
 if exists("b:is_bash")
  syn cluster shCommandSubList add=bashSpecialVariables,bashStatement