No. The fibres that matter are far too small to see individually and asbestos has no smell and no taste. There is no sensory warning of any kind, which is exactly why the whole subject is managed on paper: registers, analysis certificates, air tests and clearance procedures exist because a person standing in a room has no way of knowing what is in the air.
Why you cannot see them
The respirable fibres are measured in micrometres and are invisible without a microscope. Visible dust in a disturbed room tells you the material has been broken and tells you nothing about fibre concentration; a room that looks clean can still be above a clearance indicator, which is why clearance is a measured test rather than a look.
Why you cannot smell them
Asbestos is a mineral fibre with no odour. Any smell during work is coming from something else, typically the bitumen, the adhesive, the paint or the heat being applied. Judging safety by smell is how people conclude a room is fine when it has not been tested.
How long fibres stay airborne
Respirable fibres are light and settle very slowly, and they are readily disturbed back into the air by movement, sweeping or an ordinary vacuum cleaner. That is why an area is kept closed after a disturbance rather than aired for an hour, why clearance testing is done under defined conditions, and why the answer to a spill is to leave it alone and call somebody.
Does asbestos burn
No, and that is why it was used. Asbestos is fire resistant, which is what put it into fire protection, flues, textiles and insulation. In a fire the surrounding materials burn away and the asbestos is left, often damaged and friable, which is why fire damaged buildings are surveyed before anyone works in them.
Questions people ask about can you see or smell asbestos fibres in the air
Would I know if I had been exposed?
No, and this site does not answer health questions. If you are worried about exposure, speak to your GP or occupational health service rather than to a contractor.
Can an air test tell me the room is safe?
An air test tells you what the concentration was at that time under that sampling method. That is why clearance is a four stage procedure including a visual inspection rather than a single number.
Should I vacuum after a disturbance?
No. Ordinary vacuum cleaners pass fine fibres straight through and spread them. Leave the debris, close the area and get advice.