Living in Singapore now for quite some time, I started to think about things that I found a bit strange and/or confusing at first, as I didn't know about them, or because they just were totally different from 'back home in Germany'.  Over time, I got used to them, and I'd even go as far as saying that they're 'part of my everyday life' now.

1. The Yellow Line
   In the MRT stations, there's always a yellow line close to the edge of the platform. The yellow line indicates which distance passengers should keep from the tracks and thus the train in order to avoid any accidents.
   
   (Picture taken by and copyright by siriuslim/Mervyn Lim)
   And it works! People respect that line, you can rarely see anyone crossing it, even when there is no train arriving.
   
2. Bubble Tea
   Some years agoin the 80s, Bubble Tea (or also: Pearl Tea) has become very popular in Taiwan and has then spread to many parts of the world, including Singapore. Basically, what you get is some tea (usually milk tea) with some 'pearls' (seen here at the bottom of the drink):
   
   While I didn't like the 'pearls' at the beginning (it's a bit weird to drink something through a straw and suddenly have some small ball in your mouth), I started to like it more and more over time
   http://en.wikipedia.org/wiki/Bubble_tea
   
3. Takeaway Drink in a Plastic Bag
   One thing that I thought of as "dangerous", "risky", and "doesn't really make sense??!!" is putting a drink into a plastic bag with a straw:
   
   This is actually quite common practice if you buy a drink at a hawker centre for takeaway. And yes, I got used to it and don't mind buying it this way, though I still think that cups are way more practical (this is the cheaper solution, that's why it's in use)
   
4. Love for Escalators
   This is a very common situation that I see nearly everyday inside the MRT stations:
   A lot of people crowd around the 'entrance point' of an escalator, but ignore the stairs right next to it (or nearby). It looks approximately like this:
   
   People here prefer to wait for a short while and take the escalator instead of walking up- or downstairs. Not all, of course Especially in the evenings, when people are on their way home, there are a lot more who take the stairs and walk downstairs.
   
5. Multi-lingual announcements in the MRT stations and trains
   Being a multi-cultural society, Singapore has got 4 different languages in use: Chinese (Mandarin), Malay, Tamil (Indian dialect), and English (to make communication between people of different races easier; basically everyone speaks it). Thus, while on the MRT, announcements can be heard in all 4 of them.
   For example, the following announcement about the Yellow Line can be heard in the stations:
   

   For your safety, please stand behind the yellow line.　Thank You (English)
   各位搭客请注意，为了您的安全，请站在黄线后面。 (Chinese)
   Sila ambil perhatian. Demi keselamatan anda, harap berdiri di belakang garisan kuning. (Malay)
   அன்பு கூர்ந்து பயணிகள் கவனிக்கவும் உங்கலின் பாதுகாபை முன்னிட்டு அன்பு மஞல் கோட்டுக்கு பின்னால் நிலுஙள். (Tamil)

   
   More examples:
   http://en.wikipedia.org/wiki/Safety_on_the_Mass_Rapid_Transit_(Singapore)#Safety_announcements_and_the_yellow_line
   
6. Food Centre of the World
   Chicken Rice. Mee Goreng. Nasi Lemak. Laksa. Roti Prata. Murtabak. Hokkien Mee. Mee Soto. Nasi Briyani. Fish Bee Hoon. Wanton Mee. Yong Tau Fu. Satay. Carrot Cake. Kaya Toast. Fish Head Curry. Kuih. Black Pepper Crab. Curry Puff. Chilli Crab.
   Those are just some of the dishes you can get here. Singapore basically hosts all kinds of cuisines: Chinese (even there are some differences, e.g. Sichuan cuisine, Cantonese cuisine, ...), Malay, Indian, Vietnamese, Indonesian, Japanese, Korean, Western (generally speaking), German, Italian, French, American (well, there's Botak Jones!), Spanish, Arabian, Turkish, Argentinian, ....
   And if there's one thing that every Singaporean enjoys, then it's eating. Where else in the world would that many people drive half-way over the island in the middle of the night to get the best [insert food name from above]
   
7. Reclaimed Land
   Singapore has grown a lot, not just in terms of people, GDP, or others, but also when it comes to its actual land size. This is due to 'land reclamation', meaning that new land has been created at what was previously sea.
   The following image gives an idea what has been reclaimed during the last decades:
   
   (Note: Image taken from http://www.collate.org/collected/?p=136)
8. NEWater
   Purified wastewater (sewage....). Yes, you can drink it.
   While this may not sound very good initially, the water reclaimed through this way is actually of very high quality. The majority of the water reclaimed this way is used for industrial purposes, but a small percentage is meant for drinking.
   The reasons why Singapore started to 'recycle' water is that most of the water here in Singapore is actually brought in from Malaysia (there are giant pipelines coming in from Malaysia to Singapore) for a extremely cheap price (negotiated many decades ago), however the contract will not run forever, and new sources had to be investigated.
   http://en.wikipedia.org/wiki/NEWater
   
9. Karang Guni
   Living in a HDB, one sound that you hear very regularly (daily sometimes, even on Sunday mornings.....) is a man walking around outside with a horn, shouting something that you as a Westerner can't understand (it's usually a mix of Malay & Hokkien, as far as I know).
   What they do is collect old newspapers and other unwanted items, and sell them elsewhere and/or recycle them.
   
10. HDBs
    HDB stands for Housing Development Board, the local governmental institution that governs the public housing market. HDB also refers to the type of public housing that they manage. Whereas in other countries such housing is usually meant for people who need social support, this is a slightly different case in Singapore. Around 80% of the population live in such buildings: