Ooooh, I love this question! There are a number, but here are the ones that come pretty immediately to mind:
- Hal and Falstaff pretending to be Hal’s father, in Henry IV Part 1
- Lady Macbeth washing her hands
- Not a scene, but Macbeth’s “tomorrow and tomorrow and tomorrow” speech is possibly my favorite passage ever written
- Richard II’s return to England (”For God’s sake, let us sit upon the ground and tell sad stories of the death of kings; how some have been deposed; some slain in war, some haunted by the ghosts they have deposed; all murder’d: for within the hollow crown that rounds the mortal temples of a king keeps Death his court.”)
- Benedick thinking Beatrice is in love with him
- Helena and Hermia fighting, in Midsummer Night’s Dream
- Many of the scenes between Celia and Rosalind, in As You Like It
I’m sure there’s a slew of others I’m missing, but those are some of my all-time favorite scenes.
For performing a stand-alone scene, I should note again that I’m not an actor, so I can’t give much advice from that angle. However, this is what I would say:
Pick a scene that is especially emotive, or highly comedic. You have less time to make the audience feel something, so pick from highlights.
If you’re able to make tweaks to the scene, that’s worth considering, as well. Sometimes just clipping a few lines here and there can make it easier to pull the scene out of the context. However, that doesn’t mean you don’t need to understand the context. Read the whole play and understand your character outside of the confines of that single scene. That can go a long way.
One of the more difficult things I think about stand-alone scenes is that there is no time for the characters to be properly introduced enough to make the audience care about them. Comb the beginning of your scene for anything that can introduce your character to the audience and grab their attention. If you don’t find any in the text, brainstorm some nonverbal action that can take place on stage before the first line. What can you tell the audience about your character using only physical actions? Fidgeting, posture, physical reaction when they see another character, etc. For example, let’s say they are expecting someone and anticipating a difficult/awkward conversation. Are they nervous? Comically practicing how they will react when said someone enters? Bored? Impatient? Playing a video game on the sofa while they wait? If a coin, for instance, features later in the scene, utilize that. These kinds of nonverbal intros before the text of the scene begins need only last a few seconds, but they can go a long way to helping your audience understand your character before you ever open your mouth, and making them care about your character, too. Don’t be afraid to let this nonverbal intro go on a little bit longer than may feel comfortable. It can really set you up for success for the rest of your scene.
Similarly, don’t be afraid to use silence even after your scene begins. It’s so easy to just plow through the rest (and that may be what the scene calls for), but it can really benefit situations like a stand-alone scene to slow down a bit. It lets you communicate more, which allows the short performance to feel more complete.
I hope that helps!