Have not heard either of these buds but I have experience with and have owned buds fro
both Shure and Monster. This will boil down to what sound signature you are looking for.
The Shures will most certainly sound a lot more neutral, and will probably be very clinical and flat. These are studio-class monitors and are designed more for recording and monitoring than listening to music. One thing to note is these Shure are high impedance - 44ohms. Not that bad but these weren't meant to be driven by an iPod.
The Beats Solo on the other hand are designed to be used from an iPod. Not sure why it's impedance is but it should drive fine by an iPod. It is also I think Twice as expensive as the SRH440. It also will undoubtedly have a verybass heavy signature with a big mid range hump. It's ideal for music like RNB, trance, hip hop, house, etc.
Personally I hate all things Shure so I would cast my vote to the Dre's. I have the Monster Turbines and while I find it too expensive for what I got, they are good buds and I would take them over any Shure IEM, even the famous SE530.