Your Tiny Core Linux install probably has a bare-minimum shell installed. This is probably busybox. You can verify this by checking if /bin/sh is symlinked to something else:
ls -l /bin/sh
lrwxrwxrwx 1 root root 7 Dec 15 20:52 /bin/sh -> busybox
Since Tiny Core Linux loads the bare-minimum OS into RAM, this is desirable. However, if you need a more fully-featured shell you may install it via the Application Browser (See the List of available applications here, the command-line version of the application browser is tce-ab). Tiny Core Linux currently has: bash and tcsh.
To install bash:
tce-load -wi bash.tcz
The default shell is still /bin/sh, however you can spawn a bash shell by running bash from within it.
To set the tcuser's default shell:
which bash | sudo tee -a /etc/shells > /dev/null
sudo sed -i "/$(cat /etc/sysconfig/tcuser)/ s#:[^:]\+\$#:$(which bash)#" /etc/passwd
You might want to also install the GNU coreutils if you're looking for more GNU standard utilities for use in bash scripts also:
tce-load -wi coreutils.tcz