Can you sneak into Premium Economy Class?

I've been contracting for a major Australian airline. I've seen the app the crew uses. It has a manifest of every passenger in every class, names and notes. They most certainly know which seats are supposed to be empty, sorry ;)

(also it might cause a note to go against your name once they spot you)

It may be possible if your are lucky, but don't count on it.

As already answered by Zach, airlines very well do know who should sit where. The question is if they always check, and in which detail.

Case in point: A budget airline I travelled with recently has different "classes" even within economy. Specifically, bringing checked baggage and getting a snack are extra. We booked late and got seats in different rows. After boarding was complete, I noticed the seat next to my wife was empty, so apparently the person who had booked it didn't show up. I changed to there in order to sit next to my partner. 20 minutes later, stewardesses go through the cabin with a paper and hand out snacks to selected passengers. We booked without (weekend trip, hand luggage was enough), but I received a snack, most likely because I was sitting in a seat that had booked the higher package. They didn't check boarding passes or anything, just seat numbers.

So if you manage to sneak into a seat that was book but ends up empty, you may have a reasonable chance of not being noticed. The crew will see the seat is booked, see that someone is sitting there, and if the gender matches, will probably not question you.

They don't keep a name-list or seat-list Premium Economy right?

Wrong.

It's worth watching what the cabin staff does and talks about during the boarding and take-off preparation. One of the things that you can't overlook when you sit in front of a (smaller) plane is passing the passenger manifest by the ground staff to the on-board crew. This is pre-requisite to complete boarding. It might be handled through apps in some airlines these days but it has to be passed and confirmed for the plane to close its doors.

Then the crew discuss what kind of "special" passengers do they have and how many of them. Usually premium is handled by a different part of a crew, so they'll be fully aware which seats should be empty. I remember overhearing a discussion like "we have two infants, 5 premium and 3 business on-board today" when I was sitting in the back of a plane.

Finally, the premium economy usually has some kind of complimentary meals even on a short-haul flights. On long-haul the meal offered is of a better kind than that served in Economy. Those are also carefully counted so it will be obvious that something is wrong.

I have also personally witnessed once a person to be pushed from a premium to economy seat (she was evidently trying to "upgrade" her flight the way you describe) and once a person being requested to get back to theirs seat within Economy. So apparently crews are cautious about such things.

All the flights I'm referring here happened within Europe, on-board of different airlines though.

Several years ago on United, I asked a flight attendant if I could move from my full row to the empty middle row behind me. Apologetically, she said the crew tries to keep that row for themselves, but I could move forward to a less-full row (she specified which one). It was only after I was seated in that new row that I realized it was a minor upgrade.

A different flight attendant "caught" me in the row, made me move out of it, and was rather nasty to me for the rest of the flight.