A mutation is most often likely to be either neutral (no effect) or destabiling (protein misfolds). Changes in activity require changes at the active site or the entranceway to it —which are often only a handful of residues, so are very uncommon.
So the answer to your question is most mutations keep the activity, although a large fraction break it. While to get a new activity the process is more fluid and is not boolean.
NB. Not all proteins are enzymes, but for simplicity I will concentrate on enzymes.
Neutrality and core vs. surface
The outer residues of a protein, the surface residues, are very tolerant to mutations. Unless they are part of an binding surface (interface) or the change results in a charge change that is. The core residues on the other hand are more sensitive to mutations.
One can do a scan in silico or in vitro or in vivo to determine the mutational landscape of a protein, which can take various forms.
This being Bioinformatics stack overflow I'll give an in silico example. The following for example is a heatmap I did some time back with Rosetta pmut_scan, which is a classic tool for this and shows in the y axis the residue it was mutated to and on the x axis the position. A larger change in Gibbs free energy of folding) relative to wild type (∆∆G) means it is destabiling, but anything less than 5 kcal/mol is neutral. As you can see many are neutral, but some half are destabiling —with a caveat for proline, which require a backbone change.
In the in vitro setting one speaks of melting temperature (DSC assays) or turnover number over Michaelis constant (enzyme specific assay). In the in vivo setting one speaks of a fitness cost of a mutation, i.e. the protein is less active so the organism grows less.
Drugs inhibit enzymes by binding the active site. Therefore resistance is often costly. There are some nice papers that look into the fitness cost and mutational landscape of HIV protease or β-lactamase. In this situation a strong epistasis is seen. That is, a first mutation that confers resistance to the drug greatly destabilises the protein, while a second compensates for the first mutation.
Change in function
An enzyme can catalyse a reaction well on a given substrate, because it was evolved this way (technically the encoding gene evolved, but shh). This is the physiological activity. However, enzymes can also catalyse side activities on similar substrates with a lower turnover and higher Michaelis constant. This is especially true for substrates it was not selected against. This is called "enzyme promiscuity". The tolerance of the physiological activity to mutation is called "robustness", while the ability to increase promiscuous activities is "plasticity". Different enzymes have a different balance of the two.
So a mutation may change the strength of the physiological activity, but may result in a new promiscuous activity that can be used as a stepping stone for selection to increase this activity. Therefore, as you can see it is not a binary switch, except for a few very specific case.
Best amino acid for changes?
Proline is a very strange amino acid as it is a secondary amino acid. It lacks a hydrogen on the backbone nitrogen so it cannot form the required hydrogen bond for an α-helix. And can adopt a cis-conformation with minimal penalty (whereas it is about 5 kcal/mol for other chiral amino acids). So changes to this amino acid are often very destabilising.
Protein cores tend to be hydrophobic so changes to the hydrophobic residues are often deleterious, especially when going from a small residue to a larger one.
The remaining residues are frequently found on the surface so may be neutral. However, some are found in the active site and may be able to shift the parameters of the ligand space for that enzyme. A change in charge may result in an increased specificity for a different set of substrates for example. A size change from a larger amino acid to a smaller one may result in a broader range of promiscuous activities —mutations to alanine for example.
So ironically, polar and charged residues as they are frequently found on surfaces are the residues which are most often neutral, but in a small amount of cases they may change activity.