<form action="/update" method="POST" accept-charset="UTF-8">
...
<button type="button" class="btn btn-success"><a href="/" class="menu">Обновить</a></button>
На это: <button type="submit" class="btn btn-success">Обновить</button>
public function update()
{
foreach (request()->multipleselect1 as $value)
echo "{$value}<br>";
foreach (request()->multipleselect2 as $value)
echo "{$value}<br>";
foreach (request()->multipleselect3 as $value)
echo "{$value}<br>";
foreach (request()->multipleselect4 as $value)
echo "{$value}<br>";
}
<div class="col-lg-10">
<select class="form-control" multiple="multiple" id="multipleselect1[]" name="multipleselect1[]">
<option value="1">Овечка 1</option>
<option value="2">Овечка 2</option>
</select>
</div>
alert($("#multipleselect1[] option:selected").val());