Hi Caitiona,

The word you ar elooking for is: fret saw

Simple as that.

Regards

Ernst offers a variety of terms:

Stichsäge f / compass saw [with open handle], pad o. keyhole saw, piercing saw, sabre saw
feine Stichsäge / sash saw

The following reference includes illustrations which may be helpful:
Compass Saw
Compass Saw and Interchangeable Blades
Also Known As:
Keyhole saw, nesting saw (erroneous--not the same item)
Description:
Curved handle with long, thin, pointed blade. Most models come with a few interchangeable blades. Similar to a drywall or wallboard saw and the Japanese Keyhole Saw, but finer-toothed. Smaller versions are called keyhole saws and have finer teeth, which can usually cut metal as well as wood. Also very useful are still smaller models with metal, open construction, pistol grip or turret-head handles, which can hold the smaller blades at various angles. All of these saws are often sold with several different-sized blades, and are therefore sometimes called "nesting saws'' or "nest of saws.'' When the handle is in line with the small blade, it may be called a jab, or pad saw, as well as a keyhole saw. The term "compass'' generally refers to the smaller blade.
Use:
Cutting holes and curves.
www.manslife.com/hardmach/fixit/lightfixtures/05_06.html

Tools Dictionary - Keyhole Saw
Used for cutting openings in the middle of wood trim, such as baseboard, paneling or wood lath
www.dow.com/Styrofoam/na/dow_home/project/keyhole.htm

A sabre saw, also known as a jigsaw, is great for cutting curves and circles. But ...
www.todayshomeowner.com/repair/20010207_ss1.html

... Piercing Saw. Adjustable frame with a bow depth of 55mm. Rigid and well balanced with a hardwood handle. The blade is tensioned as it is clamped and the ...
www.gemcraft.cossaxbiz.com/product-Jewellery-Making Tools a...

For illustrations of various sabre saw blades, see
www.hawera.com/stichsaege/stichsaege.htm

HTH