I've just read the official communication from the European Commission on EU passenger rights during the coronavirus crisis and think the information will be very helpful here since I have been seeing quite a few questions popping up on it and no good source of information.
The full communication can be found here in English: https://ec.europa.eu/transport/sites/transport/files/legislation/c20201830_en.pdf
The parts I think will be of most interest to the most people are those concerning cancellations by the airline, cancellations by the passengers themselves and compensation for delays and cancellations.
Flight Cancellations by Airline
In the case of a flight cancellation by the airlines (no matter what the cause is), Article 5 obliges the operating air carrier to offer the passengers the choice among:
a) reimbursement (refund);
b) re-routing at the earliest opportunity, or
c) re-routing at a later date at the passenger's convenience.
The guidance also makes very clear that if an airline offers a voucher instead of the choice between refund or re-routing, this does not affect the passenger's right to opt for a refund.
Flight Cancellation by Passenger
In cases where a passenger cannot or no longer wants to travel, whether or not the passenger is refunded or re-routed depends on the ticket type as specified by the airline's terms and conditions.
Right to 261/2004 Compensation
Airlines, as well as a lot of the travel community, will say a passenger is not entitled to EU261 compensation for delays or cancellations related to the pandemic. This is not necessarily true: the specific cancellation/delay must still actually be beyond the control of the carrier. For example, a business decision by an airline to reduce/cancel a service for profitability reasons does NOT constitute "extraordinary circumstances".
If one suspects an airline is trying to illegally weasel their way out of paying, the national enforcement body (NEB) of the departure country should be consulted. If the departure is from a non-EU/EFTA state which doesn't have a NEB dealing with EC261 cases, the NEB of the arrival country should be consulted.
Furthermore, even when the airline is legitimately exempt from paying compensation, it still has a duty of care to their passengers who are travelling, which you can read more about.