CSS3 animation transition: opacity not working

You can just use visibility without using display:

.top-nav li ul.drop-down {
    list-style-type: none;
    position: absolute;
    top: 32px;
    left: -40px;
    visibility: hidden; 
    opacity: 0;
    -webkit-transition: opacity 0.3s;
    -moz-transition: opacity 0.3s;
    -o-transition: opacity 0.3s;
    transition: opacity 0.3s;
}

.top-nav li:hover ul.drop-down {    
    visibility: visible;
    opacity: 1; 
}

Demo.

You can use css animation when you want to increase opacity and make the element visible, like --

span {
   display: none;
   opacity: 0;
}

To make span visible

span {
  display: block;
  animation: visible 2s;   
}

@keyframes visible {
 from {
   opacity: 0;
 }
 to {
   opacity: 1;
 }
}

And if you're getting a bug when make animation less than 1s then use animation: visible 0.5s forwards;