Is it a must to buy 1 lot (100 units) or can someone buy less than 100 units due to insufficient capital to invest?

The number of lots depends on the share price, so you may not necessarily get 100 share with $500 (think DBS/OCBC). However you can buy odd lots if you really need to, but I highly discourage it. Brokerage houses will allow transaction of odd lots (I'm from Phillip and I know for a fact it can be done) but poor liquidty and wide spreads should be enough reason for you not to buy odd lots.

Rather than trying to invest in shares with $500 a month, save your funds and wait for the right price. You'll get a better deal that way, with a better margin of safety. Due to transaction costs involved, I won't recommend transacting in shares if the transaction cost is 0.5% of the contract value. If your budget is low, you would be better off doing UTs on POEMS in this case, as there are no sales charges, platform, or switching fees.

If you are buying through a brokerage house then the minimum shares you have to buy is 100 shares.

Assuming you can only invest $500 every month, buying odd lots is only possible with regular savings plan.

You can check out POSB InvestSaver, OCBC Blue Chip Investment Plan or Philips Share Builder Plan.