I tried some Dollar Store "Heavy Duty" batteries in my Time Ranger once, and it wouldn't even power up. It wasn't because the batteries were weak, but because the voltage was actually too high. HD batteries are common carbon/zinc types and a fresh one will have a voltage of over 10 volts. Most modern detectors use a voltage regulator to reduce the working voltage to something closer to 5 volts. The c/z batts voltage was higher than the regulator wanted to digest.
While the voltage might be initially high with c/z types, it quickly drops with any large current demand, and the end result is the same, incorrect voltage for the detector to work properly. Some detectors, particularly older ones, have no problem with either c/z or alkaline types. My original Gold Bug has no problem using carbon/zinc, alkaline, ni-cad or NimH types. My Falcon pinpointer will run a couple seasons on those cheapy dollar store types, so your mileage may vary.
Generally speaking, rechargeable types will work, but you can expect shorter life before recharging is needed as compared to alkalines. Some people like the new lithium batteries. They are more expensive and are not rechargeable, but do have a much longer life than even alkalines.
Nowadays, there are higher voltage and capacity rechargeable types being offered for sale, these are preferable to the older ni-cads that only provided 7.4v instead of the 9v provided by alkalines.
Physical size is also an issue to consider. Many newer batteries are just a tiny bit larger than the older ones. My original Gold Bug has a very snug-fitting battery compartment and some batteries are to large to fit without binding or getting stuck in there. I add a little grabber-tag of scotch tape to help extract tight-fitting cells.
So there's no clear and fast rule to apply in every case, but generally, the c/z types have been long outmoded, while most any alkaline or lithium, rechargeable ni-cad or NimH type should work fine.
-Ed