Capturing multiple line output into a Bash variable

Actually, RESULT contains what you want — to demonstrate:

echo "$RESULT"

What you show is what you get from:

echo $RESULT

As noted in the comments, the difference is that (1) the double-quoted version of the variable (echo "$RESULT") preserves internal spacing of the value exactly as it is represented in the variable — newlines, tabs, multiple blanks and all — whereas (2) the unquoted version (echo $RESULT) replaces each sequence of one or more blanks, tabs and newlines with a single space. Thus (1) preserves the shape of the input variable, whereas (2) creates a potentially very long single line of output with 'words' separated by single spaces (where a 'word' is a sequence of non-whitespace characters; there needn't be any alphanumerics in any of the words).

Another pitfall with this is that command substitution — $() — strips trailing newlines. Probably not always important, but if you really want to preserve exactly what was output, you'll have to use another line and some quoting:

RESULTX="$(./myscript; echo x)"
RESULT="${RESULTX%x}"

This is especially important if you want to handle all possible filenames (to avoid undefined behavior like operating on the wrong file).

In case that you're interested in specific lines, use a result-array:

declare RESULT=($(./myscript))  # (..) = array
echo "First line: ${RESULT[0]}"
echo "Second line: ${RESULT[1]}"
echo "N-th line: ${RESULT[N]}"

In addition to the answer given by @l0b0 I just had the situation where I needed to both keep any trailing newlines output by the script and check the script's return code. And the problem with l0b0's answer is that the 'echo x' was resetting $? back to zero... so I managed to come up with this very cunning solution:

RESULTX="$(./myscript; echo x$?)"
RETURNCODE=${RESULTX##*x}
RESULT="${RESULTX%x*}"