Currently, it is possible to copy selected text to a clipboard by tapping “Copy” on the context menu, but it seems that it is impossible to paste from it on iSH.

I believe both of them are essential. Besides, keyboard shortcuts like CMD+V are also welcome if possible.
Pasting is indeed impossible, mostly because I'm not sure what the best UI would be for it. Long press maybe? Or a button in the toolbar?
I prefer a button in inputAccessoryView or the standard keyboard’s paste button like the following image.

In the other hand, for instance, Blink has a “Paste” item in the context menu. It’s a little bit weird, but it works.

Looking at the blink source code, a custom popup menu doesn't seem too hard. I've added a task list to the issue.
I got the context menu working.
Hardware keyboard shortcut apparently comes for free with context menu support.
Context menu is great, though very difficult to bring up without a keyboard. I agree with what @kkk669 requested, if possible.
In the latest beta I am no longer able to copy after selecting some text.
I have the same issue, only a “paste” option appears.
I guess because WebKit is no longer the first responder. I'll have to implement -copy myself, which shouldn't be hard.
In the latest TestFlight build 35, when I just long press text to select it on iSH, it will be copied into the pasteboard even if I don’t tap the “Copy” item. Is this a bug?
I’m using 3rd generation 12.9” iPad Pro with iOS 12.1.1.
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@kkk669 I cannot reproduce it.. I am iOS 12.1 tho, it might be a small bug with iOS 12.1.1
@ELChris414 I also reproduced it on iPad Air 2 w/ iOS 12.1 and iPhone XS with iOS 12.1.1. So, it’s not iOS 12.1.1 bug.
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How to reproduce it: https://youtu.be/OoDOArAkhSg
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Saw the video, still cannot reproduce. This is weird. It just keeps pasting my last copied, it does not renew the copy
I can replicate this on iOS 12.1.1. Please, create a new issue for this. Thanks
@Mnpn03 Thanks. I created #174.
There's now a paste button on the toolbar, so I'll close this.
Now that iOS 13.7 has introduced a copy/paste dialog within the touch-keyboardless full screen view of the scrollback, I totally forgot about iSH’s button and was absentmindedly trying to paste by using the floating dialog. It is possible, but very tricky; the kb has to stay hidden, which it thoroughly dislikes.
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Looking at the blink source code, a custom popup menu doesn't seem too hard. I've added a task list to the issue.