As mentioned above, go to eBay and get a Saleae 24 MHz 8 channel clone. This will set you back about $10. Use either Saleae or Open Source software. They both have pros and cons, you will need to play with them and see which one you like. For me, Saleae is better.

Do you mean logic analysers or debuggers?

Logic analysers are devices that look at the physical traces, and you can use it with any kind of signals. Arm/MIPS/Computer serial line, whatever.

For starters, you can easily find stuff on AliExpress for ten bucks or less, and they are often "good enough". Often marked as Saleae clones, but they are completely different internally, and work with Sigrock open source software.

They will last you long enough, and for almost no money. By the time you outgrow it, you'll have the cash for a real Saleae, which you should do because they are niiiiiiiice.

This is 3x the budget range you were looking for, but from my experience, it has been well worth the money.  

 http://www.dreamsourcelab.com/order.html

Short answer is: there isn't such a thing as an ARM specific logic analyzer anymore.

Long answer: Back in the day, when micros still relied on external databuses for ROM and RAM, it was common to have architecture specific bus decoders on logic analyzers.  These features let the logic analyzer disassemble instructions as they went across the bus.  Most modern SoCs keep their memory accesses on chip and have core features that decouple memory fetches from execution, so these sorts of analyzer features aren't that helpful anymore.