A week or two after the Lunar New Year (Chinese New Year) as that when the majority of companies pay their bonuses (auspicious red packets). Once the employees get that in their grubby mitts, they scamper off to new jobs. A lot of them do, anyway, obviously not all. but had to add the disclaimer before somebody becomes aggrieved that I said all Singaporeans are job hoppers. (Just 87.63% actually)
You can try in January but they can't "take a visa for you" You can apply for a SVP (Bog standard Tourist Visa) which will give you 30 days, but you cannot work on that. You can, however go to job interviews while on it. To get a working visa you need to have a job offer. The employer has to apply for it. You cannot. Only the company can and you can only work for THAT company. If you want to change jobs, your new employer has to apply for a new working pass. As soon as you leave the employer, your pass is canceled.
Start here: http://www.mom.gov.sg/