Whether your landlord or you do the TM30 - and you can do it yourself in CM Immigration, make sure you get a printed copy of the receipt. If the landlord does it online, get a printout of the confirmation. CM Immigration always looks for the TM30 slip even thought they could, if they wanted to, look it up on a computer.
And as noted in other comments above, CM Immigration has fined people for not having a TM30 on file before. Irrelevant of some of the comments above about how it should all be on the landlord, CM immigration doesn't care. If your landlord doesn't do it, do it yourself.