Approach #1: Thanks for the Salvation & Atonement, Hashem! I Love It!
For example, the Bitachon Weekly for Parshat Shemini related an anecdote about a tzaddik named Rav Avraham Yitzchak Kohn who lived in Yerushalayim.
When Rav Kohn faced different routes to get from one place to another, he chose the route that led him past his antagonists, who then yelled at & made fun of him.
Ironically, Rav Kohn took pleasure in this public humiliation & degradation.
He knew that the searing trial of bizayonot (humiliations; degradations) rescues a person from terrible suffering, both in This World and the Next.
The people who heap scorn & humiliation on others are absolutely not right.
But if it does happen, then Rav Kohn's example is one to take heart from.
Approach #2: Hey, At Least the Fools are Enjoying Themselves!
As Rav Shteinman passed by a place where some boisterous young fellows standing outside started screaming at and ridiculing him, the men accompanying the rav wanted to disperse the obnoxious group.
But Rav Shteinman stopped them.
"Let the boys enjoy themselves," he said. "They appear to have pleasure in making fun of me; let them continue."
Let's take deeper look into this intriguing response.
Certainly, Rav Shteinman knows Jewish Law soundly condemns such disrespectful behavior.
If anyone would have asked Rav Shteinman if such behavior is allowed, he would have condemned it in the strongest terms.
But for himself, being a genuine talmid chacham, he realized the value of bizayonot.
And he decided to focus on the pleasure of his fellow Jews.
Now, the kind of person who behaves like the obnoxious fellows above generally acts out of a rowdy & uppity enjoyment of such behavior.
Again, it's severely wrong for them to act this way.
Just the fact that they derive so much pleasure from ganging up against an elderly man to verbally abuse him shows how degraded they are on the inside.
They evidence a disturbing lack of shame, not only to behave with such obnoxious disrespect, but to do so publicly, exhibiting to all exactly how deplorable they really are.
Yet Rav Shteinman focused on the pleasure (however twisted) they received from their despicable behavior.
Self-Experimentation: An Attempt at Imitating Rav Shteinman
That's why the Torah outright forbids onaat devarim (verbal abuse) or causing others any kind of pain—emotionally or mentally or physically (unless it's somehow truly beneficial to the other person, like medical treatment or to stop someone from harming himself with a bad decision or action).
That's why Judaism views humiliation as a form of murder.
Again, the Torah views mocking or degrading or abusing another person as absolutely awful.
In other words: God Himself condemns such behavior.
...if you find yourself on the receiving end of bizayon, the above examples offer helpful ways to handle it.
And no, the above are not just for "very great people."
After reading the above, I found myself in a mildly embarrassing incident during which I felt others were smirking at me.
Despite the mildness of it (and even the lack of certainty regarding whether anyone was actually smirking at me), it really stung.
But Hashem made me remember Rav Shteinman's example, and I did my best to mentally re-orient myself to focus on the idea that I was giving them enjoyment (if indeed the were smirking at me)—even if it's a very debased level of enjoyment.
And it really did take the sting away. Also, it amused me to do it. (Responding like this to bizayon is an act of absurdity, if you think about it. You could see something like this happening in a comedy scene.)
The feeling of amusement also helped dilute the feelings of emotional discomfort.
So yeah, even if you can't emulate this wholeheartedly (like I couldn't), it is still definitely helpful.
Other times, I struggled to respond to obvious bizayon with appreciation & simcha, but found that very challenging.
Despite knowing what our Sages say about the value of bizayon, I still find myself doing what I can to avoid it!
Just being honest about where I'm really holding (i.e., a fairly mediocre level).
Nonetheless, it helps to know you definitely receive reward for responding to bizayon with silent joy, appreciation, humor, or altruism.
Remember, that searing sting of bizayon prevents all kinds of terrible suffering.
While I personally am not on the level to actively seek out bizayon (because despite my intellectual understanding of its value, experiencing bizayon still makes me cringe), we definitely benefit from just the act of trying to shift our perspective to that of Chazal—as exemplified by Rav Kohn & Rav Shteinman.
What Chazal Say about Bizayon
"...that if people would know the benefit of bizayonot, they would seek this out in the marketplaces and the streets."
And if you had eyes of intelligence, you would seek out someone to bother you...
Seriously. You can plunge yourself straight to Gehinnom just for that.
On the other hand, we should strive to respond to bizayon according the principles laid out above.
(Bizayon's power of atonement & salvation derives davka from the fact that it really is so dreadfully painful.)
May we all please receive the siyata d'Shmaya we need to succeed!