"Shove" suggests quite a rough movement, so we would only use it with close family and friends. 'Shove down' and 'shove over' are very informal ways of asking someone to make room for you, on a bench or sofa. You would never use these phrases in formal situations. In fact, we wouldn't even use them neutral situations: you wouldn't say them to a neighbour, an elderly or distant relative, or to a colleague you didn't know well, for example. It would sound rude and insulting to say 'Shove down' or 'Shove over' to someone you weren't on close terms with. 'Could you move over (a bit), please?' might be a more appropriate thing to say in semi-formal or neutral situations. With strangers, you probably don't need to say very much at all, other than 'Excuse me'. A polite 'Excuse me', combined with a glance at the park bench or plane seat would be enough.

If you want something that works among friends or strangers, you could also say something casual like "Mind if I squeeze in here?" or "Is this space free?". Telling somebody to "scooch over" can work among friends, but I could also imagine somebody politely asking "could you scooch over a bit?". Maybe it's a cultural thing, but even a polite "excuse me" in this case, could sound a bit cold and impolite to me. But it really depends on the context - e.g., if you're crowding their "personal space" or they're just in the way of your rightfully designated seat.

In a casual manner you could say...Could you please move over? or Could you please scoot over? In a more formal situation you should say...Excuse me, would you be able to make some room for me?