Hi Yan Izz,
A house, in Singapore can refer to both HDB or private property.
(1) You can buy a private property with one name only, not an issue.
 (2) You can buy HDB flat as long as you fulfill the eligibility criteria. You have to be a Singapore Citizen and at or above 35yo, to buy BTO (2-rm flexi only), or any resale flat type.
Hope the above answer to your main concerns, but if there are more queries, please feel free to contact me at 
90110636 
, or email: ling.ck7@gmail.com if more information is needed.
 I'll be glad to assist.
Best regards
 Ling CK 
90110636 
 ling.ck7@gmail.com
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