Greatest common substring
Brachylog (v2), 3 9 bytes

{sᵛ}ᶠlᵒtw


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Full program. Input from standard input (as a JSON-style list of strings), output to standard output.

Explanation

{sᵛ}ᶠlᵒtw
 s         Find a substring
  ᵛ          of every element {of the input}; the same one for each
{  }ᶠ      Convert generator to list
     lᵒt   Take list element with maximum length
        w  Output it


Apparently, the tiebreak order on s is not what it is in nearly everything else in Brachylog, so we need to manually override it to produce the longest output. (That's a bit frustrating: four extra characters for the override, plus two grouping characters because Brachylog doesn't parse two metapredicates in a row.)

Brachylog's s doesn't return empty substrings, so we need a bit of a trick to get around that: instead of making a function submission (which is what's normally done), we write a full program, outputting to standard output. That way, if there's a common substring, we just output it, and we're done. If there isn't a common substring, the program errors out – but it still prints nothing to standard output, thus it outputs the null string as intended.