是的，一般使馆要求，除非在寒暑假期间，因学校放假，无法开具学校证明。可以用复印件代替。除此之外应该正常提供在校证明的，尤其是大学生。如果是和父母一同前往父母给出资的情况下，如果实在没有在校证明，只能交复印件，这样递交试试，但是很可能直接影响签证结果。建议还是开具学校证明，和复印件一并作为签证材料递交稳妥！

需要，资助证明，关心谁出钱

按澳大利亚要求提供签证资料

肯定要在读证明

美国的中国留学生在美国办理新加坡旅游签证

非常想要帮助你，但我去查了新加坡外交部官网，新加坡驻美国的领事馆相关具体信息，确实有提到需要有新加坡本地担保人。
具体通过哪个城市的大使馆自己亲自去申请，还是通过邮寄个人资料的方式申请，就看你自己了。但是所需要的资料都是如下：
1）Completed Form 14A
2）One recent passport-sized colour photograph (to be pasted on the top right-hand corner of Form 14A ) which meets the following requirements:
-Image must be taken within the last 3 months
-Photograph should be in colour, must be taken against a white background with a matt or semi-matt finish
-Image must show the full face and without headgear (headgear worn in accordance with religious or racial customs is acceptable but must not hide the facial features)
3）Please provide a day time contact number, email address and state the type of visa required – single, double or multiple journeys.
4）Original passport and one photocopy of the applicant’s passport bio-data page and the US visa page. Passport must be valid for at least 6 months beyond the last day of intended stay in Singapore.
5）Completed Form V39A (Letter of Introduction for Visa Application)
6）A photocopy of confirmed onward air ticket or flight itinerary. A valid visa to the next destination should be obtained (if applicable). If you are returning to USA after visiting Singapore, please produce a valid US visa or relevant documentary proof confirming your returnability, eg. INS Parole Paper, etc.
7）Letter of Employment and a clear copy of US Resident Alien/PR/Green Card or US visa.
8）If you are applying for a business visa, a Letter of Invitation from a Singapore registered company is required. (Letters of Invitation from US companies are not acceptable.)
9）If you are applying for a social visit visa, a copy of the local contact’s Singapore Identity Card will suffice.

相信你的英文水平肯定不需要我翻译吧。
我给你解释两个关键的表格，也就是上面的1）和5）。
Form 14A是签证申请表；而Form V39A (Letter of Introduction for Visa Application)就是你提到的介绍信，准确的说是本地人担保，具体文件是这个：%20Services/FormV39A.pdf

从官方信息得知，你如果要通过大使馆自己申请就需要在新加坡的人（公民或者绿卡）给你担保。

另外的方法就是建议你咨询你那边的旅行社或者华人旅行社，他们肯定有相关的经验，或者说应该是可以帮你办理新加坡签证的。