Edit menu parent element

For the top most ul output by wp_nav_menu(), you can change the entire wrap of of the list items (ul or otherwise) with items_wrap arg.

wp_nav_menu( array(
    ...
    'walker'  => new Your_Nav_Walker_Class(),
    'items_wrap' => '<ul class="your-class" data-u-want="some-data-you-want">%3$s</ul>',
    ...
));

User example on Codex

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For uls output from within nav walker (sub level menus, etc.), you can use something like:

function start_lvl( &$output, $depth = 0, $args = array() ) {
    $indent = str_repeat("\t", $depth);
    $output .= "\n$indent\n";
}
function end_lvl( &$output, $depth = 0, $args = array() ) {
    $indent = str_repeat("\t", $depth);
    $output .= "$indent\n";
}

Altering $depth to target submenus at different levels.

Nav_Walker boilerplate on github