Has anybody got a typical routine for a five and half month old?
Your baby, your family, is unique. Just as other families are. You would be best to avoid seeking The Definitive Answer from elsewhere, instead work out a routine that works for you and your family.
Its a massive cliché, but babies don't come with a manual for good reason - they are all different.
I have school runs to do, which therefore means I couldn't be too 'go with the flow' with baby because she had to fit in with the comings and goings of the rest of the family routine.
Once my DD started extending her naps and moved away from short, frequent naps she settled on a morning nap starting when we got back from school - say 9.00/9.30. I would then wake her up at 11am if she wasn't awake.
Others wouldn't dream of waking a sleeping baby but this is where the fact that all familys are different comes in. I know that I wanted DD to have her afternoon nap in the cot. This meant she had to be awake for 3pm (to go on school run). Working backwards this therefore meant she needed to go to sleep by 1pm to allow for a 2 hour nap.
Again working backwards from this, to be awake long enough to be ready to go to sleep again at 1pm means I have to wake my DD at 11am if she isn't awake.
So in summery, the sleep routine that works for my baby and my family at 5.5 months was:
7.00 wake
9.00/9.30 sleep in cot (if home) or carseat (if out)
11.00 woken up (will wake sooner of we are out)
1.00 sleep in cot (always stay home for this nap)
3.00 woken up
7.00 bedtime in cot
However, this works for my family. You just need to work out what will work for your family.
Are you using a dummy for sleep times? Great as a way of allowing independent sleeping.