In ivy mode, commands that have input as prefix of their name should be presented first

[rudi]

Using standard completion or icomplete-mode, M-x man RET executes the man command. When turning on ivy-mode, man is somewhere on the 8th screen. (Entering ^man on the prompt helps a bit, but once the library is loaded, various commands starting with uppercase "Man-" prefix are presented before man even in that case.) Other commands have the same problem, e.g., gnus is presented after diary-from-outlook-gnus when typing M-x gnus.

Would it be possible to sort the list of completions so that prefix matches of the input string come before other possibilities?

[abo-abo]

You should use a dedicated command for M-x:

(global-set-key (kbd "M-x") 'counsel-M-x)

It already comes with ^ as the initial input, so the issue disappears.

Alternatively, you can rely on flx to do the prefix sorting:

(setq ivy-re-builders-alist
      '((t . ivy--regex-fuzzy)))

[abo-abo]

Use:

(setq ivy-prefix-sort t)

to turn on the new behavior. It's off by default, because it can be potentially more time-consuming.
If it turns out to be fast enough, ivy-prefix-sort will be made on by default.

[abo-abo]

I reconsidered the customization interface because of #269. You should now use:

(setq ivy-sort-matches-functions-alist
      '((t . ivy--prefix-sort)))

to turn on the prefix sorting. It's more flexible, since it allows to turn on extra sorting only for some commands if you want.