⦿ have at it
⦿ knock yourself out
1 The expressions have at it and knock yourself out basically mean the same thing. They're often used when someone passes on something and gives another person permission to have it — often in reference to food or drink, but not exclusively. It can refer to anything: a job, a turn, or even an item in a store, for instance. The speaker, in most cases, is passing it up because they don't like it, don't want it, it's too expensive, they don't like the taste, etc.
Ronnie and his brother, Danny, walk into the city's new ice-cream parlor boasting 20 "unique" flavors.
______________________
Danny: Damn, this place is fire.
Ronnie: Yep, nice colors! But look at those flavors... Strawberry & Peppercorn? Avocado & Pretzel? Really?
Danny: You’re too picky. [to the server] Can I taste the Avocado & Pretzel, please?
Ronnie: And I’ll try the… what the hell — Strawberry & Peppercorn. [takes a taste] Ugh, that is disgusting!
Danny: Like I said… too picky. Here, let me try it.
Ronnie: Have at it!
Danny: [gives it a taste] Nope, you’re right. It tastes like crap.
2 Have at it and knock yourself out are also used to give someone permission to try their hand at something that you haven't been able to do yourself, or don't want to take on because it's too difficult, unpleasant, tedious, etc. We use them in situations where the speaker gives up trying to do something daunting and lets someone else give it a shot. The speaker's attitude can range from «Oh good, now I don't have to do it», to one of «Okay, I give up. If you think you can do it better than me, go right ahead and try!»
Nina comes home from IKEA with her dream bed. Her roommate, Billy, comes home to find her sitting on the floor surrounded by bed parts.
____________
Billy: Yikes, I see you have your work cut out for you.
Nina: This is ridiculous. It took me a half hour just to make sure all the parts are there. And these instructions… nothing but pictures!
Billy: [looking at the instructions] Hmmm, this looks easy enough. I could put this together in like 20 minutes .
Nina: Thank you, but I won’t have you putting my bed together on your Saturday off.
Billy: Suit yourself. Let me know if you change your mind.
[An hour later, Billy comes back]
Billy: What? You haven’t even put the legs on.
Nina: I just can’t figure anything out from these stupid drawings. Why don’t they just write the instructions out like everyone else?!
Billy: Nina, I’m an engineer. This’ll take me all of 20 minutes.
Nina: Have at it, then. I’m about to burst into tears.
3 These expressions basically also mean «go ahead» and are used to give someone permission to do something that would bring them a lot of pleasure. For instance, in this dialogue, the boss knows that Andrey would like nothing more than to get back at the office bully for the way he's treated him, so when the opportunity arises, his boss gives him full license to do exactly that.
Andy goes to his boss with proof that the office bully, David, has been stealing from the company.
____________Andy: Mr. Clark, I need to talk to you.
Boss: Come into my office. What's up?
Andy: It's about David.
Boss: Again? Honestly! Can't you just ignore him?
Andy: You're going to want to hear this.
Boss: I'm all ears.
Andy: Yesterday, when I was leaving work, I saw David selling someone shoes — our shoes — out of the trunk of his car.
Boss: That's a strong accusation, Andy. Do you have proof?
Andy: As a matter of fact I do. I recorded the whole thing with my phone.
[He shows Mr. Clark the video]
Boss: Well, this is obviously going to be Mr. Lee's last day of work and I'm going to have to report this to the police. Can you call him in here so I can tell him?
Andy: Mr. Clark, would you mind if I gave him the bad news... in front of everybody?
Boss: You know what, after everything that idiot's put you through, you've earned it. Knock yourself out!