Here is a short list of movies that mention Singapore in some way via character dialogue.

“Welcome to Singapore”

Son Of Kong
1933
‘Helstrom’ says “I heard about you in Singapore. How you captured the biggest animal on earth and taken him back to New York.”

Road to Singapore
1940

The Scarlet Clue
1945
“I remember you since last time I arrest you. Is six years ago in Singapore, remember?”

Unknown Island
1948
“Say, where did they come from? I’ve never seen them in Singapore before.
Carol, you’ve been pretty wonderful through all this. We get back to Singapore, we’ll…”

A Town Like Alice
1956
A newly wealthy English woman returns to Malaya to build a well for the villagers who helped her during war. Singapore is mentioned on numerous occasions.

Billy Liar
1963
“This mate of mine fetched them from Singapore.”

The Wild Party
1975
“Singapore Sally lived in an alley / Down by Maleno Bay/ The story of her life, her trouble and her strife / She happened to tell me one day… / Sing… sing… Singapore Sally, life gave her quite a shove. / Sing… sing… Singapore Sally, she was waiting for her long lost love!
Hey, Queenie, how about “Singapore Sally?” Me and Oskie know it!
I’m a little rusty, but Singapore Sally it is.”

Saint Jack
1979
A movie filmed entirely in Singapore – Thanks Wee Loon for the update

The Good Father
1985
“Compromise? She’s taking him to Australia. What sort of compromise do you suggest, Singapore?”

Cocktail
1988
“I am the last barman poet / I see America drinking the fabulous cocktails I make / Americans getting stinky on something I stir or shake / The sex on the beach / The schnapps made from peach / The velvet hammer / The Alabama slammer. / I make things with juice and froth / The pink squirrel / The three-toed sloth. / I make drinks so sweet and snazzy / The iced tea / The kamakazi / The orgasm / The death spasm / The Singapore sling / The dingaling. / America you’ve just been devoted to every flavor I got / But if you want to got loaded / Why don’t you just order a shot? / Bar is open.”

Romper Stomper
1992
‘Sonny Jim’ says “I vote we get a 45 automatic and blow the head **** back to Singapore.”

Johnny Mnemonic
1995
Keanu Reaves mentions that his implants are from Singapore.

Fear and Loathing in Las Vegas
1998
“Never mind. Let’s get right to the heart of this thing. Twenty-four hours ago we were sitting in the Pogo Lounge of the Beverly Heights Hotel, in the patio section of course, drinking Singapore Slings with mescal on the side, hiding from the brutish realities of this foul year of Our Lord, nineteen-hundred and seventy one…”

Office Space
1999
‘Bob’ says “Oh yeah, we’re gonna bring in some entry-level graduates, farm some work out to Singapore, that’s the usual deal.”

Rogue Trader
1999
“No laughing matter. Outraging a lady’s modesty, very serious crime in Singapore.”

Catch Me if You Can
2002
Tom Hanks’ character ‘Carl Hanratty’ lists of countries including Singapore: “South America, Australia, Singapore, Egypt.”

Pirates of the Caribbean: Curse Of The Black Pearl
2003
‘Jack Sparrow’ says “Clearly you’ve never been to Singapore.”

King Kong
2005
‘Hayes’ says “If someone were to tell you this ship was headed for Singapore, what would you say?”

Batman Begins
2005
One of the places where Wayne Enterprises transports Batman’s equipment is through Singapore.

Pirates of the Caribbean: The World’s End
2007
Chow Yuen Fatt welcomes the pirates to Singapore.

The Amazing Spider-Man 2
2014
Peter Parker mentions Singapore when speaking of Electro’s evolution to Harry Osborn

Hopefully I can add some more…

13/9/2014 Thanks Angeline for the following additions:
A Town Like Alice
Catch Me if You Can

Thanks Tristan for:
Road to Singapore