If you purchased the flat from HDB directly or had applied for the CPF housing grant
>> you need to satisfy the minimum occupation period of 5 years before you may purchase private property.
However, if you had purchased the flat from the open market without CPF Housing Grant, you are able to purchase private property (for investment).
In both situations, if you do not sell your HDB, you will need to stay in the HDB flat.
There are no additional fees incurred.
There is a point to note that property tax for investment properties or properties that are not owner occupied will be liable for a higher property tax.
Also note that per property - other taxes like TV and Radio Licence fee is payable per property you own.
Hence before you plunge into investing in private property - contact me for a consultation.