There is an unofficial prayer area on level 2 near to Pepper Plus (#02-117,).

From Harbourfront MRT (NE1/CC29), 
1. Walk to Exit B, leading to the escalator to Harbourfront Centre Level 1.
2. Once you've entered Harbourfront Centre, take the escalator up to Level 2 and look for the escalator leading to Unity (#02-118/119). 
3. Once you're in front of Unity, walk towards Pepper Plus (#02-117,) then turn right and walk in the direction of the toilets. On the way to the toilets, on your right (past the glass doors but before the lift lobby), you will see a discreet corner which is the unofficial prayer area. 
The unofficial prayer area is shared for muslimah and muslimin. Qibla is available. Please bring your own telekung and prayer mat. You may perform your ablution at the toilets just past the lift lobby. 

Alternatively, if you don't mind walking to the next nearest musollah (apx 10-15 mins), there is one at Vivocity. You can find general directions here. 
P.S. There is a link at level 2 between Vivocity & Harbourfront Centre.

There is an unofficial prayer area on level 2. Please note that you will need to walk down 2 flight of stairs. 
1. Upon walking in via the main entrance, turn left and you will see an escalator leading to level 2.
2. Take the escalator up on level 2 and then turn right till you see the toilets. You may perform your ablution here. 
3. After performing your ablution, walk straight and turn right towards My Little Campus (#02-344). You will see an Exit Door in between the 2 My Little Campuses. Walk towards that Exit Door (It is on the second floor just above KAAAKIS #01-342)
4. Push the Exit Door and walk down 2 flight of stairs to the first level. Just before the other Exit door, on the right is the unofficial prayer area which is shared for muslimah and muslimin. Prayer mat and qibla are provided. Please bring your own telekung. Please hang the prayer mat after use. After praying, you can either go back up the stairs or alternatively, you may open the Exit Door to you left which will lead you to the front of the main entrance (near to Koufu). 

There is a musollah for muslimah and muslimin available for patrons. For assistance, please ask the Muslim staff. 
The musollah is in the middle of the food court, near to Power Thai BBQ stall.
The musollah is separated by a partition and prayer mats are provided. Please bring your own telekung. Ablution can be performed at the toilet nearby.