In each set of 5, one of the words is an "odd one out". The odd ones out, in
each column, occur in alphabetical order, just like the sets of five (except
for the one at the bottom of the fourth column, where we accidentally swapped
two). One of those is an odd one out, as well.

Here are the properties we were looking for, and the word that did not match:

== Column 1 ==
Letters in reverse alphabetical order (BROWS)
Forms a word when you prepend S (FED)
Names of 80s bands (HAMBURG)
Start with a negative prefix, though they have no positive form (NUMB)
Alternate consonants and vowels (QUART)

Column: You can add -ER to make a new word (FED)

== Column 2 ==
Have two meanings with different pronunciations (ERA)
Anagrams of ANGRIEST (FRIGHTEN)
Contain three consecutive letters of the alphabet in a row (MENTIONED)
Anagrams of days or months (NETWORK)
Words that mean something different in British English (WEIGHTY)

Column: Contain numbers (TEN, ONE, TWO, EIGHT) as a substring (ERA)

== Column 3 ==
Groups of animals (CHOICE)
All three vowels are the same (COOKIE)
Contain four of the same letter (DECODE)
Start and end with the same three letters (KICKED)
Contain adjacent pairs of double letters (LAIR)

Column: Horizontally symmetrical words (LAIR)

== Column 4 ==
Contain names of programming languages (DELIVER)
End with names of NFL teams (DESSERTS)
Adjectives that describe themselves (DRAB)
Contain one vowel (GATEMAN)
"Self-antonyms": words with two meanings that are roughly opposite (MARS)

Column: Spell another word backwards (MARS)

== Column 5 ==
Don't rhyme with anything (ADDED)
Form another word when you remove the first letter (BEEF)
Are their own plural (CABBAGE)
Contain all five vowels (DEFACED)
Words made up by The Simpsons (HALS)

Column: Words that can be spelled with musical notes, as in A-G (HALS)

The odd words out are FED, ERA, LAIR, MARS, and HALS. These letters exactly
fill the blanks below, spelling FEDERAL AIR MARSHALS.