Solutions

ACROSS
1 Select.
S(E+L)ECT
4 Carpet.
CAR(P+E)T
9 Best.
(two meanings)
10 Inhabiting.
I+N+H+A+BIT+IN+G
11 Averse.
A+VERSE
12 Turnover.
TURN+OVER
13 Estimated.
ESTIM+A+TED (times anag.)
15 Ally.
ALL+Y
16 Nips.
(anag.)
17 Better off.
BETTER+OFF
21 Research.
RE+SEARCH
22 Fuming.
F+U+M+IN+G
24 Allegiance.
ALL(EG)IANCE
25 Chat.
C+HAT
26 Setter.
(two meanings)
27 Manage.
MAN+AGE


DOWN
1 Sheaves.
S+HEAVES
2 Later.
LATE+R
3 Chimera.
CH+IM+ERA
5 Auburn.
AU+BURN
6 Pot-boiler.
POT+BOILER
7 Tannery.
T(ANNE)RY
8 White elephant.
WHIT+E+ELEPHANT (with anag.)
14 Implement.
(two meanings)
16 Needles.
NE(E)D+LES
18 Taffeta.
TAFF(y)+ETA
19 Fanfare.
FAN+FARE
20 Praise.
P+RAISE
23 Mecca.
ME(C)CA (came anag.)