Tree root emerging from lawn. What to do

[NewspaperTaxis]

I'm trying to tart up a long neglected lawn, though I'm leaving it late in the year! However, I noticed a big root emerging from the surface and I'm not sure what to do about it. In one case there isn't an important tree nearby so maybe I can just hack it out and there won't be a 'Timber!' moment where it crashes down.
Another root on another part of the lawn is a bit larger and near a big tree in the neighbour's garden.
These roots were not here decades ago - it's the family pile - so what should I do? They're too obtrusive to just chuck some soil over and hope the grass grows over it.

We’ve got several large trees round our house. We had roots this large and larger cut out for an extension (by a man with a digger) and the trees survived.

I’ve cut out large roots I’ve found in the new borders with a pad saw and we have some that just get run over by the mower. Could you create a border around them?

They're not big, the tree can survive just fine without them. Dig down a bit and cut them off horizontally with a saw just below ground level.

Okay, thanks I'll give it a go!