As per the PHB, a character who uses Wild Shape can benefit from your class features (e.g. multi-classing into Rogue too) as long as "the new form is physically capable of doing so". In the answers below, I have only included features available to a Rogue up to level 18.

I think most of the core Rogue advancement features would work as usual:

• Expertise - works as normal (you retain all skill and saving throw proficiencies in beast form; if the beast's bonus is higher use that one instead PHB p.67)
• Cunning Action - works as normal (nothing preventing a beast from using its bonus action to Dash, Disengage or Hide; or Aim included in Unearthed Arcana: Class Feature Variants)
• Uncanny Dodge - works as normal (nothing preventing a beast from using a reaction or benefiting from damage reduction)
• Evasion - works as normal (nothing preventing a beast from benefiting from the damage reduction for a damage source with an area-of-effect)
• Reliable Talent - works as normal (you retain all skill and saving throw proficiencies in beast form; if the beast's bonus is higher use that one instead PHB p.67)
• Blindsense - works as normal (nothing preventing a beast from benefiting from blindsense. If the beast already has this feature, the benefits do not stack; use the sensory radius which reaches farthest)
• Slippery Mind - works as normal (you retain all skill and saving throw proficiencies in beast form; if the beast's bonus is higher use that one instead PHB p.67)
• Elusive - works as normal (a beast can benefit from attack rolls not having advantage against it)

The exceptions would be:

• Thieve's Cant: Obviously reading or deciphering something would be possible in most scenarios, but not conveying something back.
• Sneak Attack: is out of the question in most cases because "the attack must use a finesse or a ranged weapon" (PHB, p.96). The possible exception could be wildshaping into an Ape (MM, p.317) as they do have a Rock ranged attack; possibly at the DM's discretion an Ape could wield a weapon with the finesse descriptor and still be able to use Sneak Attack. (see: Can a druid in Ape wildshape use weapons?)

From the Roguish Archetypes in the Player's Handbook and Xanathar's Guide to Everything, this is what I gather:

THIEF:

• Fast Hands - not likely to work (most beasts are not able to use, so it would not be possible to use Cunning Action and Thieves' Tools to disarm a trap or open a lock; might be possible in Ape form at the DM's discretion)
• Second-Story Work - works as normal (if the wildshape can climb, then climbing does not cost any extra movement)
• Supreme Sneak - works as normal (a beast can benefit from having advantage on a Dexterity (Stealth) check)
• Use Magic Device - may work sometimes at DM's discretion (see: Can a familiar attune and wear or use a magic item?)
• Thief's Reflexes - works as normal (nothing prevents a beast from benefiting from an extra turn)

ASSASSIN:

• Bonus Proficiencies (disguise kit and poisoner's kit) - unlikely (it requires being able to use a kit, but may be possible in an Ape form at the DM's discretion)
• Assassinate - works as normal (a beast can benefit from advantage on an attack roll and can deal critical damage)
• Infiltration Expertise - irrelevant (a beast cannot pass for a humanoid identity)
• Impostor - irrelevant (a beast cannot mimic humanoid speech, writing or behaviour to create an indiscernible ruse)
• Death Strike - works as normal (a beast can attack a creature that is surprised)

ARCANE TRICKSTER:

• Spellcasting - does not work ("You can't cast spells" while wildshaped, but spells requiring concentration can continue to be active in Wild Shape as long as they are cast before transforming (PHB p.67). Druids do gain access to Beast Spells at 18th level and can cast spells that do not require a material component while wildshaped. However, that build is not possible in RAW as it would require 21 character levels: Druid 18; Rogue 3.)
• Mage Hand Legerdemain - only works if Mage Hand is cast before transformation
• Magical Ambush - does not work (you cannot cast spells in beast form)
• Versatile Trickster - only works if Mage Hand is cast before transformation
• Spell Thief - works in part (a beast can use a reaction so you can still force a creature to make a saving throw and negate the spell on a fail; you still know that spell for 8 hours and the creature is not able to cast that spell - however you would need to come out of Wild Shape to be able to cast the spell yourself)

INQUISITIVE:

• Ear for Deceit - works as normal (a beast can make Wisdom (Insight) check)
• Eye for Detail - works as normal (a beast can make a Wisdom (Perception) and/or Intelligence (Investigation) check)
• Steady Eye - works as normal (a beast can move up to half speed and benefit from advantage on Perception and Investigation checks)
• Unerring Eye - works as normal (a beast has senses and can benefit from this feature as long as it is not blinded or deafened)
• Eye for Weakness - does not work as requires Sneak Attack (possible exception as above, with an Ape's Rock attack)

MASTERMIND:

• Master of Intrigue - mostly not relevant (Most beast forms would not be able to use a disguise kit, forgery kit or a gaming set; at the DM's discretion it might be possible in Ape or other forms. The additional languages would remain active, but it would not be possible to communicate while in wildshaped. (PHB, p.67))
• Master of Tactics - works as normal (there is nothing preventing a beast from using a bonus action or the Help action)
• Insightful Manipulator - works as normal (a beast can observe a creature for a minute and thus benefit from this feature)
• Misdirection - works as normal (a beast can be the target of an attack and use its reaction)
• Soul of Deceit - works as normal (the druid/rogue will still have thoughts while wildshaped and thus be able to conceal these if so wished)

SCOUT:

• Skirmisher - works as normal (nothing prevents a beast from using its reaction and moving up to half its speed, so it can benefit from this feature)
• Survivalist - works as normal (nothing prevents a beast from benefiting from double proficiency bonuses when using Nature and Survival skills)
• Superior Mobility - works as normal, mostly (a beast can benefit from an increase of 10 feet to their walking, swimming or climbing speed; however, a creature's flying or burrowing speed does not benefit from this because Superior Mobility is explicit about the types of movement which are affected)
• Ambush Master - works as normal (a beast has an initiative roll and can make attacks)
• Sudden Strike - does not work as requires Sneak Attack (possible exception as above, with an Ape's Rock attack)

SWASHBUCKLER:

• Fancy Footwork - works as normal (a beast can make a melee attack and move in its turn, so it can benefit from this feature)
• Rakish Audacity - works in part (a best could grant itself an initiative bonus; but the additional way to use Sneak Attack would not work - the possible exception as above, with an Ape's Rock attack)
• Panache - situational (it works only if there is a "shared language", i.e. with another similar beast)
• Elegant Maneuver - works as normal (there is nothing preventing a beast from benefiting from advantage on Dexterity (Acrobatics) or Strength (Athletics) checks)
• Master Duelist - works as normal (a beast is able to make an attack roll, so it can benefit from this feature)

Although I have done most of the footwork on this answer, I would like to acknowledge Nautarch and Sarcastic Villain for their helpful suggestions and for contributing to the overall understanding of how Wild Shape features in this multi-class build.