Change of mind of a job offer

You will only be blacklisted if the company lodges a complaint against you.
If you followed proper procedure and did not break the rules in your employment contract (especially about starting date and termination procedures), they should not complain. But it is better to follow up and check!

If you have an issue of only one week (assume you are providing correct information) then not sure why company is not willing to extend your joining date. Secondly, signing a job contract is not give you rights to work in Singapore, rather approval of work pass. There are instances, work pass is pending more than 3 months, during that period neither company can force you to work nor you will be allowed to work here.

So, discuss politely and request your issues with them. If there is no end result, then join on date mentioned in your offer letter (if and only if your work pass gets approved) or else anyway you can wait for your visa approval, which may take beyond few weeks (assume your joining date is less than 15 days from now, you want to extend another 7 days). Good luck

How can I check myself if I'm blacklisted from MOM?

The company really don't allow me to extend 7 days even I give them a proper explain. As in from Penang malaysia. I will need a month notice with my current company and I have some personal issues to be settle for a few days. And moving down to Singapore I will need another few days too. I sign the offer letter on 23rd Oct and they want me to join on 27th Nov. And I was making the request to extend after a week 28th Oct. When I request to them they told me first impression last. It's a very hard to make this decision of not joining. So I told them I will reject the job.

So I don't know how will they complaint me thru MOM. They didn't give me any work pass number. If I can check thru MOM website myself with passport number?