I've been googling because my lawn is prone to being muddy because it's on clay and in some areas has I think the remains of a hard standing underneath it . ( so it needs digging up ! )
I think much will depend on why yours is muddy - poor drainage like mine or too much footfall .Or heavy rain ? Too many trees ? If it's just the mowing perhaps resseding ?
Anyway the results of my investigations are below .The advice about applying sharp sand seems a little mixed - will it hold water or will it mix into the soil and improve drainage .Widly recommended is aeration - I've never been succesful at that .It just seemed to break the grass up and leave more earth exposed .
I think lawns are very hard work .I just massively lower my expectation .
I've googled and found these bits of advice
"My solution was to apply tons - and I do mean tons, of sharp sand. I do about an inch layer twice per year. The grass has never died, but I do fast rake the whole lawn a couple of times in the following weeks. If in doubt, go for a shallower layer.
It really cuts down on all the mud mess.
Sometimes I add about 1/4 fine compost and a few buckets of blood fish n bone and on occasion add grass seed to the mix.
You have to apply the sand twice a year or so. If there is a danger you will kill off some grass, do add reliable grass seed with the blood fish n bone Grass will root anywhere, even on tarmac so don't believe people who say you need a deep tilth when patching a lawn. When adding seed I would add some fine compost as I say, just to improve chances of it taking."
" try over seeding with a tough grass and cutting your gress at a slightly higher setting around say 3-5 cm. To toughen it up even more rake the grass regularly to get rid of thatch and moss which hold the moisture and cintribute to muddy gardens imho."
" If the site does not naturally dry out, deep tine aeration is the first stage of the renovation process and then some over seeding and top dressing to re create the surface back into a lawn. The spreading of a sharp sand will assist but do not forget that sand sucks up water, often making the site muddier. You cannot work a liquified soil so patience is a great virtue. A few days of wind and sunshine will pay great dividends and grass is a tough plant that can survive a manner of things thrown at it."