Day 1 probably doesn't matter, but it would be reasonable to have your app in sometime within the first week or so, i.e. the end of the month isn't a good idea. Where this really becomes a problem for most folks is looking at your application and pulling the "well, if I only took another day or two or three I could make it better maybe..."
The other problem is that places may screen apps but won't formally evaluate them until they are complete WITH LETTERS OF RECOMMENDATION. Be warned that while many places say that they don't really look at apps until MSPEs come out, I dare say that most have already got it sorted and all the MSPEs do is serve as a double check that you weren't completely BS'ing your app. I got an offer or two before MSPEs were released.
So, takehomes:
1) get your app in as soon as is reasonable... if you find yourself making up a reason to stall, send it in that day
2) figure out a clever way to gently nudge your LoR'ers to mail their damn letters
3) don't worry about MSPEs
4) oh and finally, in case there's anyone out there who is thinking "well I'll apply to 20 places first and see how that goes and then maybe apply to more"... this may not be a good strategy and you may be better-served by applying broadly at the outset. A couple of thousand bucks now is a lot cheaper than not matching for a year.
Just my 2 cents.