Where are you? The kind of grass you have may vary a lot depending on where you are. If you've got cool season grass, it sounds a lot like most of your lawn is KBG and the wide leaf grass is tall fescue (likely K31). It's very hardy, but not as attractive in a lawn and its faster growth rate makes it stick out. If that's what you have, you could spot spray with glyphosate and if the patches are not too large, the KBG will fill in from surrounding areas. If the patches are too big for that, you could spray a few areas at a time. The problem with killing all of the K31 and overseeding is that it might be tough to match the rest of the lawn (and if you buy the seed at a "big box" store, it may well include K31).

Yellow or purple nutsedge (the only difference I'm aware of is the flower color). Pull it as soon as you see it: after just a few leaves, it grows a small tuber (the "nut"), which can sprout a new plant. (The nut is usually left in the soil when the plant is pulled.) You have a good chance of eradicating it, as you can see the color difference (some can't), and have noticed that it grows more quickly than the grass after a mowing. Keep pulling the lighter, taller bits as soon as you see them. I got it all out of my backyard, but will never be able to eradicate all the nutsedge in the pasture, so at times more pops up in the back lawn..

OKGirl, You have to keep watering in mind on this garden. I would not recommend the Miscanthus or the P 'Karley Rose'. They will do well in a regularly irrigated situation but this doesn't look like such a spot. As you drive around, you will notice that the Miscanthus that look lush and green are in areas with sprinkler systems for the most part or in well tended beds in private residences. The ones in unwatered situations (Pennisetum and Miscanthus) I see around here are stunted with curly tips and look like c--p. The Ravenne Grass is being used in Highway Plantings and other very dry situations and it looks great in isolated plantings too. Also the Chaste Trees are being used in the same way and they are very lush and have soft colors with continuous bloom making them a great grass companion. Another grass needing regular watering in my own yard is the Korean Feather Grass and the 'Karl Forrester'. The blue Fescue's don't do well here. The humidity gets them and you will probably loose them after a season or two. A better choice is Helictochon (Blue Avena Grass) for a low blue grass. Its more dramatic as well. I am considering getting rid of the rest of my 'Karly Rose'. It leans from the shade coming on late afternoon even though it gets a full amount of sun and worse, it seems to need a lot of water to keep it from looking dry. I decided I don't much like it because it looks too prissy. I think the Pink Gulf Muhly is a good choice but it only looks good planted in masses in my opinion and doesn't mix well with 'Karl Forester' or Pennisetum plus they have differing water demands. I would definitely recommend the Muhlenbergia Lindheimeri from SRG to go with it and the Little Bluestem (any flavor) and the Mexican Feather Grass. I would also consider that cultivar of Indian Grass (very blue) from SRG. You can then easily increase your planting area as time goes by with seeds as well from each of these.

I think it's usual to cut the grasses down in the spring before the new growth starts. That way you get the ornamental aspect through the winter and I think having the top on protects the plant from the weather a little.