M+S is basically what classifies as an "All Season" tire. Unless you are super specialized racing or touring tires you are very likely M+S rated. It means the tread pattern and siping meets some basic requirements.
3 Peak rated means that the tire exceeds a specific level of acceleration in snow. That's all. It's a pretty important certification and I wouldn't run a tire without it in the winter if you do a lot of wintry driving. You can find regular tires with 3 peak these days - these are often called "All Weather" (the Cross Climate is a popular all weather tire these days). Some (not all) A/T tires have 3 peak rating. To answer your question - i would expect every tire with 3 peak also has M+S, as it would be very challenging to meet the 3 Peak standard without also meeting the M+S standard.
Winter tires are the next level up, the actual tire material is softer and should not be driven in higher temperatures for long if you care about your tread life and traction. Winter tires provide significantly better ice and slush handling compared to non winter tires with 3peak or M+S. Every winter tire should have 3 peak rating as well as M+S.
Still want more? Add studs to your winter tires and destroy that road with massive grip.
MORE? Chains are very useful to carry for emergencies, even if you are already on suitable tires - in particular if you do a lot of mountainous driving in winter.
I expect that i have made things even more confusing instead of helping. SORRY!
Me - i have a RAV4 that gets changed over to Blizzaks (winter tires) for the winter season. I am trying out my 3 peak rated AT tires this winter on the cruiser. If i decide i would like better traction in the cruiser, i might do a separate winter tire set in the future for the cruiser. I do encounter pretty icy/slushy conditions very often because I have an incurable skiing addiction.
