Popularity is one thing, but a conveyancer who finds themself too busy to communicate with you, whose assistant may take over, or where the level of service you were expecting drops to an unacceptably low level, is a situation you should avoid - your property sale/purchase is too important to face such service.
But how to go about minimising the chances is harder than it looks.
This first method is quite a good test: how prompt are they when sending you their first bit of communication? For example, their quote of legal fees. How do they mean to go on through the course of the whole transaction. Slow, or prompt? If they promise an email, how prompt are they in sending it. Or do they not even offer an email? Or, if you are seeking more than one quote, by say a global email, who is first in coming back to you.
Clearly speed is not an end in itself, as mediocre can be fast/prompt. So it is just one factor.
But, whoever is first, how much effort do they expend? Do they attempt to give you any information about what service you can expect from them, who they are, their team, their firm? Do they make express assurances of quality/availability upon which you can rely? Or is their email just clinical: "here is our quote....yours sincerely"?
And then the icing on the cake, from any extra information they send, do they suggest they are expert? Will their conveyancers be head-set wearers, bespoke legal training beyond degree level or you are left not knowing at all? Ask, if in doubt.
A second useful method is to check how big the conveyancing team is. The smaller they are - say 1-2 lawyers - makes it more difficult to cope with more and more new work, same for holiday cover, office absences and training days - making availability to progress your work patchy or impossible. Alternatively, a healthy size team of multiple lawyers should offer seamless cover at all times keeping your property transaction on the move.
A third method, combined with the above two is to seek out independent signs of quality. Is their law firm Lexcel accredited? Are they members of the Conveyancing Quality Scheme (CQS)? And, have they received any Awards for their work. That is rare, so it can be quite a find, if you discover one.
(But not all awards sadly. So many are 'purchased' (i.e a law firm can only be considered if (1) they pay admission to the awards and (2) the pool of potential firms to be considered for an award, are only those who have paid.))
A final health-check, is to ask a few local - and here is the crucial factor … 'independent' - estate agents (not usual national chain estate agents) whether your choice of a particular solicitors firm is a good one. Or in fact does the estate agent suggest a particular local law firm too, also on your radar.
All, or a combination of the above, should hopefully set you towards securing a time dedicated conveyancer for your property move. Good luck.