<form action="/update" method="POST" accept-charset="UTF-8">
...<button type="button" class="btn btn-success"><a href="/" class="menu">Обновить</a></button><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());