I have just graduated and my Student Pass is expiring soon, next month, but I have not guaranteed a job.
Can I stay in Singapore when my Student Pass expires?
Any suggestion on this situation?
You are required to surrender your Student Pass when/before it expires. If you do so in-country at the ICA, you will automatically receive a short-term visit pass (= essentially tourist visa) that lets you stay on in Singapore for a while (days or weeks, depends on your nationality).
If you want to stay for a longer period (up to a year), you can try your luck applying for a Long-Term Visit Pass. Being a bona fide graduate of a Singaporean university and looking for work is one of the possible criteria for getting one.