I ran into this issue and thanks to various postings in various forums, was
able to figure out the solution but didn't see one posting that laid the
whole issue out cleanly. So thought the following might help others ...

------------------------------------------------------------------------
The Problem
You use vim to create a simple shell script in a cygwin directory but the
script doesn't run properly. Example:

A script called test.sh:
#! /bin/sh
ls
echo "++++++"
cat a1.txt

produces this when run:
$ ./test.sh
./test.sh: line 2: $'ls\r': command not found
++++++
: No such file or directory

------------------------------------------------------------------------
The Solution
The problem is that vim has written the file in "DOS mode", which puts
an extra (as far as cygwin is concerned) carriage return (aka: control-M,
^M, \r) at the end of each line.

You can see that using the "-v" option with cat:
$ cat -v test.sh
#! /bin/sh^M
ls^M
echo "++++++"^M
cat a1.txt^M

The solution is to bring the file back into vi, set vim into binary mode with
the ":se binary" command, then save the file. Now, "cat -v" shows that
the carriage returns have been removed:
$ cat -v test.sh
#! /bin/sh
ls
echo "++++++"
cat a1.txt

And the program runs just fine:
$ ./test.sh
760log2trk.pl a1.txt spot.gpxcon.pl.rc
760log2trk.pl.rc a2.txt spot.gpxcon.tailers.kml
++++++
This is a file with a
few lines of text
in it.

------------------------------------------------------------------------
Note 1: The first line of the script file is common practice for a Unix script,
but isn't really required for the script to work in cygwin.

Note 2: Running the script by putting "./" in front of it is required in
cygwin as long as you don't have "." (the current directory) in the search
path. By default, cygwin doesn't include the current directory when looking
for commands. And for security reasons, it's not a good practice to put .
in your search path.

Note 3: Having saved the file in binary mode, you don't need to worry
about that setting for future edits of the file. vim will recognize the nature
of the file when it opens it and will keep it in that mode, so lines added
to the file will not be written with the carriage return. (vim tells you
that the file is in binary mode by putting "[unix]" after the file name
at the bottom of the window.)

Wait, do you have a question, or is this a cool tip you are sharing with the world?

Nope, not a question, just a tip (or set of tips). Sorry for the confusion.