First of all, as far as I know there is no TV currently made that is 44". You can get 40", 42", and 47" panels.
Second, there's really no such thing as an LED TV. There are two kinds of panels you can currently purchase in that size: Plasma or LCD. LCD works by running a LCD filter in front of the back light. Historically, the back light has been a cold cathode fluorescent lamp (CCFL) on the edges of the display. Manufacturers are slowly converting this to LED arrays. LED back lit TV's come in two flavors: edge lit and full array.
An edge lit display works exactly the same as the old CCFL displays. It has all the downsides (hard to make good blacks, and bleeding brighter areas near the edge of the screen). The only upside is that they can make the display much thinner, lighter, and more power efficient. None of those are really worth the cost over a traditional back light though.
A full array LED back lit TV has a number of LED's directly behind the full area of the panel. Most of these have local dimming. What this means is that the array of LED is divided up into zones (typically 100-200 zones). Each zone can be independently turned on and off. This helps with blacks (if an area of the screen is black, you simply don't turn on the LED's in that zone). It also eliminates the bleeding on the edges of the screen. However, when you have one bright area on a dark screen, you can still see bleeding around it. You still get the power saving features, but you lose the thinness factor. If you buy an LED back lit LCD this is what you want.
There's lots of talk of refresh rates with LCD. This is basically meaningless. The panel can sport an insane refresh rate (480Hz) but if the individual pixels don't have a response time to keep up, motion will look blurry. 60Hz is more than enough to fool your eye as long as the pixels have a good response time. However, if you go with LCD; I would advise a 120Hz TV. The only reason for this is that films on blu-ray are traditionally 24 FPS and TV is 60 FPS. 120Hz is the minimum refresh rate that can display both of these without further processing the video. The more you process, the more quality you lose. This also means that you want to turn off any motion smoothing effects built in to the TV. What they do is guess at what would be between the two 60 FPS frames on TV or the five 24 FPS on blu-ray and show them between the actually frames that were sent to you. While this is theoretically possible, I've never seen an implementation that is actually good. If you are shopping for a TV and looking at them in the store make sure this is turned off!
All of that being said. Your best option is still Plasma. Plasma works by turning on and off individual pixels. You can think of it being like an LED back lit TV with 2,073,600 zones. Also, because each pixel can be turned on and off individually the entire screen doesn't need to be refreshed which each frame, which makes the ghosting of moving images much better.
There are a few downsides to plasma though. They will be heavier and more power hungry than an LED back lit LCD. Neither of these are serious problems. The Panasonic VIERA TC-P42S30 only weighs 62 lbs. This is just 20 lbs more than a 42" full array LED Vizio LCD (XVT3D424SV). The Panasonic is Energy Star rated, but would probably cost you $20 more in electricity over a year to run (depending on how much you run it). The TV is over $200 cheaper though, so you have to factor that in. Also plasma will never be as bright as LCD. This is really apparent when you view the pictures side by side in a very bright store like Costco or Best Buy. But unless you watch your TV in a brightly lit fluorescent room, you'll never notice this in practice. Also some LCD's have matte screens versus glossy screens found on plasmas (though this is becoming more rare). The glossy displays generally have anti-reflective coatings on them that mitigate this and as long as you aren't in a room with tons of bright windows, you won't notice it.
If I was buying a TV on a budget in the 42" range, that Panasonic would be what I went for. It's hard to find a true local dimming LCD in that size range. The Vizio I mentioned is out there, and there are a few others, but they are much more expensive. So my recommendation is the Panasonic VIERA TC-P42S30. Amazon currently has a good price, but shopping around you might beat it.
That is off course unless you need 3D, but then we need a whole other conversation.