SH adheres to an official industry standard, but it is not suitable for scientific computing. Bash is considered an informal standard (e.g., by Google). Bash 3 is preferable in most of situations in the world of bioinformatics.
As already described in other answers, SH (
/bin/sh, plain Bourne shell, the original UNIX shell) should fully adhere to POSIX which is a real industry standard. However, SH is too limited for scientific computing since many key features were incorporated later in SH successors, especially in Bash (
/bin/bash, Bourne Again Shell):
set -o pipefail,
[[ ... ]], or process substitutions
< () to name at least few.
In practice, it is much more difficult to write "safe" scripts in pure SH and only shell experts are usually capable to prevent unexpected behavior. For example, it may be hard to ensure that no command in a pipeline failed in the middle of computation. For Bash, various easy-to-follow defensive-programming recommendations have been developed and they should prevent all these problems. From this reason, many computer scientists, software engineers and companies use Bash as a kind of a standard. For instance, Google internal policy allows only Bash for writing any shell scripts.
Even though we cannot expect that Bash is present on completely every Unix machine (e.g., on mobile devices as @terdon pointed out), a vast majority of *nix machines used for scientific computation should have it. We also should be aware of the fact that Bash can be slower than SH and that it has recently suffered from major security issues. Moreover, various Bash versions exist and scripts working on modern Linux machines with Bash 4 might not work on OS X, which is still based on Bash 3.
To sum up, Bash 3 is probably be the most reasonable choice for scientific computing.
I addressed the comments from @terdon and @John Marshall. In particular, I added an explanation why Bash is more suitable for scientific computing than SH (in my opinion).