This answer addresses the initial question briefly but is added mainly to provide information on the modern equivalent to the spiral cut resistor which is being asked about. In most cases the through hole tubular resistor would not be used and an "SMD" equivalent probably would.
Question [paraphrased]: Carbon film resistors are basically an insulating (probably ceramic) rod covered in a carbon film in a spiral around it. However, this seems very similar to a coil.
Are there any situations where this kind of resistor is avoided in AC applications?
The applicable terminology for having properties like a coil are "inductive" and inductance.
As Maxthon says: Yes, they do have inductance but it is usually a minor effect compared to their resistance.
At "radio frequencies" it may matter.
Order of inductance is probably in the 10's to 100's of nanoHenry range. [This range is similar to what Spehro indicated in his answer.]
Related - the modern equivalent:
The following relates to a modern equivalent - the surface mount "chip resistor". These are available in carbon film (older) and metal oxide film (most newer ones) and also work by utilising a layer of conductor which is "fired" onto an insulating surface. They also usually have resistances which are to broadly distributed to be suitable for use directly and so they also are "trimmed". Because the resistor element is (usually) a flat rectangle the trimming does not use a spiral but instead either material is trimmed off one side of the rectangle or (more usually) a range of patterns are cut into the rectangle to increase the overall resistance. For experimental purposes (usually) the resistive layer may be abraded (eg by "sanding or scraping) to decrease its thickness and so increase its resistance.
The various methods of altering resistance have various obvious and less obvious effects on resistance characteristics. Long term resistance values, amount of "drift" and rate of drift are process dependent and to obtain high accuracy long term stable resistors significant effort may be required.
The usual zillion pictures - each picture links to a related page
'Modern" chip resistors are often LASER trimmed in various ways. Methods vary with some methos having less impact on RF performance than others.
See eg Useful document. Pictures were missing when I looked but this might be due to Win 8.1 / Boxing Day too much food blues / Murphy / reboot needed - looks like a worthwhile page if pictures appear.
Advanced resistor trimming aspects +/- 0.02% target.
- Manufacturers are now being required to shrink the case size as well as improve the gain adjustment capability, final trimmed accuracy, and repeatability. In order to accomplish these goals simultaneously and in a manner consistent with the high throughput requirements of standard TFCR production lines, a new generation of green (532nm) laser system has been developed. Unlike previous IR laser based systems, the green wavelength affords a significant reduction in the spot size of the laser and a corresponding reduction in the kerf width and thermal effects. Combined with a unique serpentine cutting routine, this provides the basis for a significant increase in the gain adjustments and an increase in final resistor accuracy and repeatability.
Use of user trimmable resistors - Vishay Why what you get may not be what you want if you do it wrong and how to do it right.
Useful 1 page resistor trimming summary This is a company's 'what we can do page' but shows offered trim patterns and achievable results.
Basic trimming cuts
At 1st glance this looks like "just" a Vishay 6 page resistor spec sheet - Post insertion trimmable with instructions but contains instructions on how to post insertion LASER trim the products.
LASER trimming machinery - awesome stuff