Just a few things to consider:
At 20 to 40 yards 6x magnification should be sufficient - the higher the magnification the more handshake becomes an issue and making out something small like a bullet hole will be more difficult. Above 10x requires a very steady hand or support.
Roof prisms make for compact binoculars but inexpensive ones are dim due to the low-quality prisms
Porro prisms are preferable but the design is less compact. High-quality prisms are BAK4. BAK7 prisms are medium-quality
You do not want an exit pupil below 3 mm for daylight use and for use in poor lighting, you want as large as you can afford or 7mm. The exit pupil also affects how easy it is to see the full field of view so even with binoculars needed for daylight use the exit pupil has relevance. The exit pupil is calculated by dividing the objective lens diameter by the magnification. For example, 7x50 binocs have a 50/7=7.1mm exit pupil.
You may prefer individual eyepiece focusing over center-focusing with diopter adjustment on one eyepiece.
You may prefer set-and-forget focus binoculars with a deep depth of field
If you want to use the binocs for hunting you may want a rangefinding reticle
You can also get monoculars with rangefinding reticle
A monocular should cost less and can be very compact.
Tasco 8x32 for just under $50 may be worth looking into. Tasco is usually decent. They do weigh 14oz, though.
At 1 ounce, this is the tiniest useful ATN monocular I own. It fits in the pocket watch pocket of jeans. It's actually quite good but went out of production a decade ago. I'm sure similar monoculars are available.