Some details I gleaned from SgWiki:
UBC No. 182 (Yellow) Upp. Bukit Timah Rd (Near Ewart Circus) -
Benoi Road (Jurong Industrial Estate)
VIA
Upp Bukit Timah Rd, Dunearn Rd, Stevens Rd, Scotts Rd, Orchard Rd,
Clemenceau Ave, Pulo Saigon Rd, Canal Rd, Upp Pickering St, South
Bridge Rd, Neil Rd, New Bridge Rd, Outram Rd, Tiong Bahru Rd,
Alexandra Rd, Ayer Rajah Rd, Upp Ayer Rajah Rd, Jln Ahmad Ibrahim,
Corporation Rd, Jln Tukang, Quality Rd, International Rd, Jln Boon
Lay, Jln Ahmad Ibrahim, Jln Besi, Liu Fang Rd, Jln Besi Baja, Jln
Besi and Benoi Rd, near Singapore Shipbuilding Engineering Co.
Ltd.
...I will spare you the return trip.
If you attempt to trace out this bus route on a map, you may think
you are drawing highly-detailed spaghetti at some parts.
A rationalisation in 1974 made the route less insane:
182 Routeings changed to following:
Via Upper Bukit Timah Road, Dunearn Road, making a 'U' turn in
front of Dunearn Road Outpatient Dispensary, Bukit Timah Road,
Stevens Road, Scotts Road, Orchard Road, Clemenceau Avenue, Pulo
Saigon Road, Canal Road, New Bridge Road, North Canal Road, South
Canal Road, Boat Quay, Market Street, Cecil Street, Boon Tat
Street, Shenton Way and loop back along Parsi Road, Anson Road,
Robinson Road, Cross Street, Upper Cross Street, Havelock Road,
Pulo Saigon Road, Clemenceau Avenue, River Valley Road, Tank Road,
Clemenceau Avenue, Penang Road, Somerset Road, Grange Road, Orchard
Boulevard, Paterson Road, Scotts Road, Stevens Road, Bukit Timah
Road and Upper Bukit Timah Road.
I wonder if the old route 182 that later became one of TIBS' early
bus routes grew out of this 182. Compare the above with this, taken
from the 1979/80 vus guide:
Service 182: Woodlands New Bridge Road
Admiralty Rd, Woodlands Town Centre Rd, Woodlands Rd, Upp Bt Timah
Rd, Jln Anak Bt, Dunearn Rd, Scotts Rd, Orchard Rd, Dhoby Ghaut,
Bras Basah Rd, North Bridge Rd, South Bridge Rd, Neil Rd and New
Bridge Rd.
New Bridge Rd, Hill St, Stamford Rd, Orchard Rd, Penang Rd,
Somerset Rd, Grange Rd, Orchard Blvd, Paterson Rd, Scotts Rd, Bt
Timah Rd, Upp Bt Timah Rd, Woodlands Rd, Woodlands Town Centre Rd
and Admiralty Rd.
I would think they cut back the bus route from Shenton Way to New
Bridge Road at the CBD end, and extended it to the then-developing
Woodlands at the Bukit Timah end, some time between 1974 and
1979.