I've tried two or three commercial software programs, which I still use even though I purchased them a decade or more ago, so be aware that what follows may be out of date.

Dart, I think it was called, was utterly useless.

DCArt can be particularly good at interpolating over vinyl scratches, though the results are often highly dependant on the start and end points chosen for the interpolation, slight changes in these causing big changes to the results. It has quite a wide range of filters, etc, and in particular can filter based on a sample, which is particularly useful for turntable rumble. However, the software is somewhat 'quirky'. In particular, as the file is edited in situ, with its time and date stamp being altered accordingly even if you undo all changes, you should always back up a file BEFORE loading it into DCArt.

SonicFoundry (as was) / Sony (now) Sound Forge is also quite good, with a good selection of filters, etc. However, be aware that this software has potentially hidden expenses - for example, the basic program may cost so much, a filter to read in MP3 files, so much extra, etc. This approach to marketing by Sony completely alienated me, which is one reason why I'm still using a decade-old version.

As for the noise itself, generally, it's better to avoid it getting into the recording in the first place, than to try to remove it afterwards, as this inevitably will affect the wanted part of the recording to a greater or lesser extent - that is it becomes a question of whether the noise in the recording is worse than the wanted part of the recording would be after treatment to remove the noise. What can be done depends very much on what sort of noise it is, and the OP is not specific about this.

Extraneous noise picked up by a microphone is very difficult or impossible to remove. I have a recording off the soundtrack of Don't Look Back of Joan Baez, then Dylan's girlfriend, singing part of the then unfinished Percy's Song in their hotel bedroom. Her rendition of it was utterly beautiful, but during it Dylan is tactlessly pounding away at a typewriter. I've tried everything I know to remove Dylan's rudeness from the recording, but without success.

Tape hiss is comprised of mostly high-frequency random noise, and many software programs have a tape-hiss filter specifically for removing this. However, it's nature means that it can not be completely removed from a recording without adversely affecting the top-end, particularly transients, etc - it's a compromise, a question of removing as much hiss as is required to reduce its prominence in the recording without audibly degrading the wanted sound. I suspect that early Fleetwood Mac CDs of their eponymous album (7599-27241-2) and its successor Rumours (7599-27313-2) were over-treated in this respect, and, compared with the original vinyls, they sound as though the speakers are behind heavy drapery - muffled and lacking in transients. By contrast, those tracks which are also on the Greatest Hits CD (7599-25838-2) sound fine on that. I have read that other masterings of those albums do not suffer from this, but not having heard them, cannot comment.

Vinyl scratches are usually of such short duration that they can be edited out of the waveform, usually and preferably by software interpolation, failing that simply by chopping them out. However, this doesn't always work, some types of music, for example pipe music, being particularly difficult to fix. Usually the best approach with the latter is to replace the affected part of the waveform with an intact identical piece of waveform from close by.

Vinyl wear and dust noise is best removed by cleaning the vinyls before recording them, but failing that, or for whatever remains, the problem becomes similar to tape hiss, and again many programs have a specific groove-noise filter. Again the trick is apply just enough to reduce the prominence of the noise without audibly affecting the wanted sound.

Turntable rumble can be tricky to remove. The approach I use is to sample it from the lead in and lead out of the vinyl, and use the sample to create a specific filter, which DCArt can do. However, as with all of these things, it's a question of how much one can reduce the prominence of the noise without audibly affecting the wanted sound.

Mains hum can be tricky to remove. If it's just a pure sinusoid at 50 or 60Hz, in principle that should be treatable by a suitable filter of that frequency, and again software programs often have such a filter, and again it's a compromise between reducing the hum and audibly affecting the rest of the sound. However, mains hum is often not a pure sine wave, but has many harmonics, etc. Many years ago I bought off Derek Brimstone a commercial audio-cassette of one of his live performances, which has quite prominent mains hum from the PA. When later I digitised it, I wasn't able to find a way of removing the hum from the wave file.