Hiya,
Was in same position as you and am doing my internship now.
Most PR companies in Singapore hire interns. Especially the international ones. Why not? They're cheap and sometimes, even free and PR companies almost always run on a relatively tight budget. However they do not advertise as there is no need too. Enough people write in on spec for them to choose from. Research the ones you want to apply to and send in your CV with a covering letter. Sounds trite, but that's what I did and how I landed a place. If you have to, call the main line and ask for the relevant person/email to email your application too. Sometimes, the email they give on their website doesn't get checked for weeks. The receptionist may also know exactly who is looking to hire and will give you their direct email. That said, BE VERY NICE to whoever answers the phone. I've heard the receptionist in my company almost refuse to give a particular caller the relevant email address because that person was v. rude.
Bear in mind that a lot of communications and business students from the local unis, SMU/NTU would be applying now or even before for their mandatory internship which starts soon in January. Places are filling up, so start applying soon!
As for journalism, dont' know what type of journalism you wish to get into but IS Magazine frequently advertises for editorial interns. You may wish to check it out. All the local female magazines also take interns and from what I heard, procedure is the same as above!
PM me if you have anyore questions and the best of luck. Once you get your internship, be prepared to swallow your pride and start doing the dirty work...
I am as strong as the friends I have, the hairspray I use and the coffee I drink.