The answer is:
CHINESE LANTERN
First:
In each group, one word is the odd one out because it doesn't fit a pattern that fits the other words.
AID AND ABET, CAT SCAN, HAIR TRIGGER, NINEVEH, SEES
SEES, each word starts with a broadway musical, AIDA, CATS, HAIR, NINE (from @HTM)
AFTEREFFECTS, FREEZER, NOCUOUS, VERTEX, WARFARE
NOCUOUS, each word can be typed with only the left hand on a keyboard (from @HTM)
ALUMINOSILICATES, BELOVED, HIMALAYAN, INDIA INK, VERISIMILITUDE
INDIA INK, each word alternate vowels with consonants
ANTHERS, DOUGLASS, FIREBAR, SLUMBER, THOUSAND
ANTHERS, each word has a construction material in its suffix, GLASS, REBAR, LUMBER, SAND
BATHROBE, BRAZEN, CAROUSEL, ELLIPSIS, WEIGHTIEST
ELLIPSIS, take away first and last letters of each word to form a new word, ATHROB, RAZE, AROUSE, EIGHTIES
CARBON COPY, CROSSTALK, PENNY DREADFUL, TROUSERS, TURN UP
TROUSERS, each phrase contains a word which clues a letter that starts the other word: CARBON = C (chemical symbol), CROSS = T (T-cross), PENNY = D (symbol for predecimal penny), TURN = U (U-turn) from (@Stiv)
CARDIOID, LINNAEAN, NICOTINIC, PALEOETHNOLOGY, TOE OF FROG
NICOTINIC, each word contains an ABA vowel pattern, cardIOId, linnAEAn, palEOEthnology, tOE Of frog
COMMENCEMENT, DETOURS, JASMINE, JOINT HEIRS, THIS
COMMENCEMENT, each word's suffix is a possessive, OURS, MINE, THEIRS, HIS
CONSOLATE, CORRIGIBLE, NEATH, SENSICAL, WHELMED
NEATH, each word is normally used with a prefix that makes it the opposite, DISCONSOLATE, INCORRIGIBLE, NONSENSICAL, UNDERWHELMED
DAGGER, EVITATIVE, LOZENGE, PILCROW, TIE
EVITATIVE, each word is a symbol
DEIFIED, KOOK, RIYAL ICON, PUT UP, RADAR
RIYAL ICON, each word is a palindrome
DIE, ERT, GOOSE, SERAPH, SURGEON GENERAL
ERT, each word is a singular whose plural is nonstandard, DICE, GEESE, SERAPHIM, SURGEONS GENERAL
HIGHLIGHTER, HOTHATCH, INDEPENDENT, PHENOMENON, SEVENTEENTH
HOTHATCH, each word contains a repeated 3 letter pattern, IGH, NDE, ENO, ENT
LABYRINTHINE, MOTHY, PATHED, THINNER, UNTHINKABLE
LABYRINTHINE, for each word, replacing the TH with an "S" sound makes a word: MOSSY, PASSED, SINNER, UNSINKABLE
Then:
The odd words out each belong to 2 other groups (except for EVITATIVE which belongs to 3). We can list these words in a cycle based on the 2 groups they belong to (from @hdsdv). If we start with COMMENCEMENT (perhaps a hint as to where we should start), we get this:
COMMENCEMENT = Repeat 3 letters (MEN), Suffix building material (CEMENT)
HOTHATCH = Suffix building material (THATCH), Cryptic (HOT = H like on a faucet)
INDIA INK = Cryptic (India = I from NATO alphabet), ABA vowel pattern (IAI)
NOCUOUS = ABA vowel (UOU), Prefixed means opposite (INNOCUOUS)
ERT = Prefixed means opposite (INERT), Typed with left hand
SEES = Typed with left hand, Palindrome
EVITATIVE = Palindrome, Musical (EVITA)
LABYRINTHINE = Musical (LABYRINTH), Possessive suffix (THINE)
ANTHERS = Possessive suffix (HERS), Replace th with s sound (ANSWERS)
NEATH = Replace th with s sound (KNEES), Remove first and last letter (EAT)
TROUSERS = Remove first and last letter (ROUSER), Irregular plural (TROUSERS)
ELLIPSIS = Irregular plural (ELLIPSES), Symbol
RIYAL ICON = Symbol, Alternating consonants/vowels
NICOTONIC = Alternating consonants/vowels, Repeat 3 letters (NIC)
Now, taking the first letter of each word, we get CHINESE LANTERN.