TL;DR, I offered and promised in the Post Request Thread a guide to the four highest value tips I know for doing public speaking. Here they are, with explanations below:

Fortissimo! Don't apologize for talking
Know the first and last line of your comment before you open your mouth
Think about speeches/comments as having a narrative arc
Look for additional emotional tones to layer on the content

My background: I was a debater in college, but not in the Gatling-gun style of competitive debate. We did philosophical debate, where you only argued for propositions you actually believed. So, style was supposed to make it easier to get interested, but not be too Dark Arts-persuasive. I coached younger members on how to present their speeches and have spent a fair amount of time murderboarding people (helping people prepare for interviews or presentation).

I think the tools in this post are useful both for speeches you prepare and polish ahead of time, but also to be better at speaking coherently off the cuff (long and short form). You can check out my speaking style here. (I'm not using notes, and I didn't memorize a speech -- I memorized an arc which gave me room to improvise). So, here are the habits that help:

1) Fortissimo! Don't apologize for talking.

In E.L. Konigsberg's About the B'nai Bagels, the protagonist is preparing for his bar mitzvah and asks his brother for advice on how to sing his Torah portion. After listening to him, his brother has the following feedback:

“I have only one word of advice to give you”

“Give already”

“That word is fortissimo… it’s Italian for loud. When in doubt, shout, that’s what I’m telling you.”

“I should shout? Everyone will hear for sure how bad I am.”

“But, my dear brother, if you sing loud and clear, it will be easier on the audience. You’re making it doubly hard on them. Hard to listen to and hard to hear.”

Not everyone needs to be louder when they speak, but a lot of people who are uncomfortable with public speaking signal that discomfort in posture or vocal tone (a lot of freshman and sophomores had an about-to-cry sounding tension in their voices when they were speaking). If you're apologizing for talking, your audience will assume there's a reason and start to resent it or feel uncomfortable.

So, don't apologize for talking. Don't start with disclaimers ("I'll be fast, I don't want to waste anyone's time"). And don't apologize with your voice or your body language. You can get specific feedback by taping yourself talking and have a friend watch it with you and have you practice standing taller or speaking a bit more intently. You can pay a theatre grad student to meet with you a couple times about posture of voice projection. You can also just consciously review why you are talking in the first place before you open your mouth, so you remember why your comment is useful and you're giving people a gift by talking, not being an imposition.

2) Know the first and last line of your comment before you open your mouth

It's pretty obvious why you want to know the first line of your speech/answer/whatever before you start talking; you don't want an awkward lag or a spot where you might panic. But most people don't plan their conclusion ahead of time (totally neglecting the peak-end rule!).

I hear a lot of novice speakers start strong, and then kind of peter out at the end of their response. Sometimes people will just trail off, hoping someone else will pick up the slack. Sometimes people have essentially already given their closing thought, but not noticed, and then they end up repeating it awkwardly.

If you know what your closing image/sentence/line/etc is when you start talking, you know what you're aiming at from the beginning, so you won't get diverted as easily. You've removed one common cause of failure/panic in speaking, so you can speak more confidently in the first place. And your point will be more memorable/easier to engage with if you have a strong conclusion.

3) Think about speeches/comments as having a narrative arc

So that's the opening and the closing of the talk, but what goes in the middle? In English class, you probably leaned this model:

Thesis
Evidence 1
Evidence 2
Evidence 3
Thesis restated

This is terribly boring and difficult for people to retain. It's a lot more fun and memorable if you can put things in the framework of a story. Here's one formula for creating a narrative structure from Miss Snark:

X is the main guy; he wants to do:

Y is the bad guy; he wants to do:

they meet at Z and all L breaks loose.

If they don’t resolve Q, then R starts and if they do it’s L squared.

When I'm teaching class, I tend to use one that's more like:

Ever notice how you always X when you'd really like to Y? So did I! I tried Z and it turned out to work, but I wasn't sure why! I poked around in the literature and found A,B, and C, which caused me to tweak my solution to Z' and now I Y all the time, and you can too!

Basically, instead of just having a point and supporting data, you take your audience through a couple emotional arcs. It's easier to remember stories than just data. It's also more fun for your listeners to repeat, so they'll get to share your idea with others. It helps you stay away from a monotone or totally even affect while speaking (more in the next tip) and keeps the structure of your comment really clear in your own head.

Planning plot summaries of my speeches means I don't need to carry notes or memorize lines, anymore than I recite funny stories I share with friends. I can just remember the outline of the story and then expand or contract individual parts depending on what the audience responds to. The structure gives me a safety net. This way, I'm not unsure what I'm saying when I open my mouth, but I'm not stuck saying specific lines.

4) Look for additional emotional tones to layer on the content

It's boring to just listen to someone explain facts. Having a narrative arc (as above) will automatically inject some variance into your tone and affect. In my teaching example above, the emotional notes look something like this:

Frustration: [Ever notice how you always X when you'd really like to Y?] Shared identity, all of us looking at the frustration together: [So did I!] I tried Z and it turned out to work, but pleased but perplexed: [I wasn't sure why!] I poked around in the literature and surprise, but increasing feeling of catharsis: [found A,B, and C, which caused me to tweak my solution to Z'] and triumph: [now I Y all the time], return of fellow feeling and pleasure at sharing something cool: [and you can too!]

But there's more you can add. One friend of mine was explaining a counterintuitive study in a fairly matter of fact way, but it was a lot more enjoyable and memorable to hear about if she shared her surprise at how it turned out. A lot of the time, it's simplest to just make sure you're letting your honest reactions to what you're saying come across.

But, if you're not sure what those are, or want to explore other options, you can try dividing what you're saying into beats. (Beats is a phrase used in theatre for subdivisions within scenes. In one conversation or story, the dominant emotional tone can change, and that transition is the start of a new beat). So, try dividing up your notes or your outline into sections and just experiment with the dominant tone for the section. Here's a reworking of the emotional beats in my teaching outline:

Sadness, regret: [Ever notice how you always X when you'd really like to Y?] Shame shared as vulnerability: [So did I!] I tried Z and it turned out to work, but tentative, a little uncertain: [I wasn't sure why!] I poked around in the literature and feeling of tinkering and assembly: [found A,B, and C, which caused me to tweak my solution to Z'] and peace, tranquility: [now I Y all the time], warmth, joy: [and you can too!]

Try looking at this list of some possible emotional tones, and see what it's like when you using them as you talk through your outline. Try reading wrong tones to a friend, to notice why they're wrong or to catch yourself if you were unnecessarily restricting your options. Sometimes tone can change a number of times in one passage (as in this marked up example), just pay attention to what prompts the shift. You can try picking a speech or a sentence that already exists, and reading it deliberately with different tones each time to get some practise and comfort using them.

So, if you work on these tips, people will be more comfortable listening to what you say (1), you'll open and close strongly (2), with a narrative arc that keeps you on track and makes your points memorable (3), and enough emotional variation to keep your audience engaged with you and your content (4). Huzzah!