Hi! My 12 weeks old has started to move in her sleep and pushes herself all the way to the side of the bassinet, with her face completely smushed against the bassinet wall (not mesh but hard). Should I move her into a cot? Is it safe for her to sleep like this? She is lying on her back and with her head looking towards the wall, completely touching it but still breathing fine.

There is no need to stop your baby from wriggling & changing her position, but it is important to make her environment safe in light of what she is doing.
Many babies are moving & rolling by 3 months of age. This is very normal.
Is she rolling to her side or wriggling & turning her head to the side?

How does she go with tummy time when awake?

Most babies will be rolling over by 4-5 months…....some earlier, some later.
This is very normal as babies grow & become stronger many babies then prefer to sleep on their tummy &/or side.
Are you still wrapping/swaddling her?
*Time to stop swaddling/wrapping if you haven’t already.
*It is important that you move her from bassinet to her cot now.
*Baby needs arms free & the space of the cot to roll safely.

Once she is rolling it is important to make the environment as safe as possible for this to occur.
Continue to place baby on the back to sleep.
Ensure cot mattress is firm, well-fitting & flat.
Ensure there is nothing loose or soft in the cot.
Preferable to use a sleeping bag when rolling that has appropriate size neck & arm holes.

Red Nose has the following information when baby is rolling.
Give baby extra tummy time to play when awake and supervised as this helps baby to develop stronger neck and upper body muscles which in turn enables them to roll back over.
Make sure that baby is on a firm and well-fitting mattress that is flat (not tilted or elevated).
Make sure that baby’s face and head remains uncovered (do not use lambswool, doonas, pillows, cot bumpers or soft toys)

As babies grow and develop they become very active and learn to roll around the cot.
At this time, continue to put them on the back at the start of sleep time, but let them find their own position of comfort.
By this stage it is not necessary to wake during the night to turn baby over to the back position

*Do not use any devices designed to keep baby in a particular sleep position.
https://rednose.com.au/article/what-do-we-do-now-that-a-baby-has-started-to-roll-over