In my experience software tools are very useful, but you can't rely on them to get rid of unwanted noise. For starters, most noise reduction software is designed to clean up music or human voice - the latter is common, and in fact for the MixPre Recorders you can actually get a recorder plug-in to de-noise interviews (i.e. to extract aircon hum etc. from human voice.
Without spending a huge amount, the 'best' noise reduction software is probably Izotope RX Studio. I have been using this, but 'gently', as I find that it does strange things to my recordings - sonograms can show pale horizontal bands post processing, and a recording I have of Hume's Warbler calls, developed a rather odd set of harmonics post processing. This was sufficient to audibly notice the difference. RX Studio is quite expensive, but they do have offers quite often when the software can be purchased with a substantial discount - but it still isn't cheap!
An alternative to consider is Steinberg's WaveLab Pro 11. Again this isn't cheap - but it is cheaper than RX. This works very differently. Basically you can take your recording and get the software to split the recording into separate layers for noise, tonals (basically whistles), and transients (basically short tics and tacs sounds). Each of these layers can then be edited visually, including reducing loudness. If you have an annoying mic pop on the transient layer, you can simple highlight it and press delete. I regularly use this software to reduce the loudness of noise, and to make adjustments - I have even toned down or deleted the buzz of a fly close to the mic and even human voice. It is not perfect, and you need to be careful again of over-processing and ruining the bird vocals.
Probably a safer way to get rid of some noise is to use a equalizer - lots of software have equalizer functions. With this I set up the equalizer to reduce the low frequency and high frequency noise, below and above the bird vocals. Most equalizers will show you the sound split over the frequency range, and the equalizer profile, so that you can visually make sure you are not impacting the bird vocals. Obviously, this works well with either low pitched or high pitched noise, but not if you have unwanted noise in the middle to the frequency band for the bird vocals. I have used this technique to significant cut out wind rumble, but it would be unsuitable for say cicada's incessant drone.
If mic pops and other unwanted transients are the problem, then a tool that allows you to do surgical modifications to the sonogram can be very useful. With this kind of tool, you can normally simply copy a small slither of sound from immediately adjacent to the unwanted transient, and paste it on top to make the unwanted noise disappear. I started out with Wavelab for this purpose, but I think it is overkill and too expensive to recommend. Cutting and pasting large areas of noise to overwrite more noisy bits of the recording rarely works, as there is normally some sound, that then gets repeated and sound wrong.
Of course the best way to get rid of noise is to get a good signal to noise ratio, by getting as close to the bird a possible. But that isn't always possible, and in our noisy world unwanted sound can still ruin a recording.
